HANDBOOK OF MATHEMATICS

I.N. Bronshtein � K.A. Semendyayev � G. Musiol � H. Muehlig

5th Ed.,With 745s Figures and 142s Tables

TABLE OF CONTENTS

1
• Preface to the Fifth English Edition 4
• From the Preface to the Fourth English Edition 4
• Co-authors 5
• 1 Arithmetic 1
2
• 1.1 Elementary Rules for Calculations 1
3
• 1.1.1 Numbers 1
4
• 1.1.1.1 Natural. Integer, and Rational Numbers 1
• 1.1.1.2 Irrational and Transcendental Numbers 2
• 1.1.1.3 Real Numbers 2
• 1.1.1.4 Continued Fractions 3
• 1.1.1.5 Commensurability 4
3
• 1.1.2 Methods for Proof 4
4
• 1.1.2.1 Direct Proof 5
• 1.1.2.2 Indirect Proof or Proof by Contradiction 5
• 1.1.2.3 Mathematical Induction 5
• 1.1.2.4 Constructive Proof 6
3
• 1.1.3 Sums and Products 6
4
• 1.1.3.1 Sums 6
• 1.1.3.2 Products 7
3
• 1.1.4 Powers, Roots, and Logarithms 7
4
• 1.1.4.1 Powers 7
• 1.1.4.2 Roots 8
• 1.1.4.3 Logarithms 9
• 1.1.4.4 Special Logarithms 9
3
• 1.1.5 Algebraic Expressions 10
4
• 1.1.5.1 Definitions 10
• 1.1.5.2 Algebraic Expressions in Detail 11
3
• 1.1.6 Integral Rational Expressions 11
4
• 1.1.6.1 Representation in Polynomial Form 11
• 1.1.6.2 Factorizing a Polynomial 11
• 1.1.6.3 Special Formulas 12
• 1.1.6.4 Binomial Theorem 12
• 1.1.6.5 Determination of the Greatest Common Divisor of Two Polynomials 14
3
• 1.1.7 Rational Expressions 14
4
• 1.1.7.1 Reducing to the Simplest Form 14
• 1.1.7.2 Determination of the Integral Rational Part 15
• 1.1.7.3 Decomposition into Partial Fractions 15
• 1.1.7.4 Transformations of Proportions 17
3
• 1.1.8 Irrational Expressions 17
2
• 1.2 Finite Series 18
3
• 1.2.1 Definition of a Finite Series 18
• 1.2.2 Arithmetic Series 18
• 1.2.3 Geometric Series 19
• 1.2.4 Special Finite Series 19
• 1.2.5 Mean Values 19
4
• 1.2.5.1 Arithmetic Mean or Arithmetic Average 19
• 1.2.5.2 Geometric Mean or Geometric Average 20
• 1.2.5.3 Harmonic Mean 20
• 1.2.5.4 Quadratic Mean 20
• 1.2.5.5 Relations Between the Means of Two Positive Values 20
3
2
• 1.3 Business Mathematics 21
3
• 1.3.1 Calculation of Interest or Percentage 21
• 1.3.2 Calculation of Compound Interest 22
4
• 1.3.2.1 Interest 22
• 1.3.2.2 Compound Interest 22
3
• 1.3.3 Amortization Calculus 23
4
• 1.3.3.1 Amortization 23
• 1.3.3.2 Equal Principal Repayments 23
• 1.3.3.3 Equal Annuities 24
3
• 1.3.4 Annuity Calculations 25
4
• 1.3.4.1 Annuities 25
• 1.3.4.2 Future Amount of an Ordinary Annuity 25
• 1.3.4.3 Balance after n Annuity Payments 25
3
• 1.3.5 Depreciation 26
2
• 1.4 Inequalities 28
3
• 1.4.1 Pure Inequalities 28
4
• 1.4.1.1 Definitions 28
• 1.4.1.2 Properties of Inequalities of Type I and II 29
3
• 1.4.2 Special Inequalities 30
4
• 1.4.2.1 Triangle Inequality for Real Numbers 30
• 1.4.2.2 Triangle Inequality for Complex Numbers 30
• 1.4.2.3 Inequalities for Absolute Values of Differences of Real and Complex Numbers 30
• 1.4.2.4 Inequality for Arithmetic and Geometric Means 30
• 1.4.2.5 Inequality for Arithmetic and Quadratic Means 30
• 1.4.2.6 Inequalities for Different Means of Real Numbers 30
• 1.4.2.7 Bernoulli's Inequality 30
• 1.4.2.8 Binomial Inequality 31
• 1.4.2.9 Cauchy Schwarz Inequality 31
• 1.4.2.10 Chebyshev Inequality 31
• 1.4.2.11 Generalized Chebyshev Inequality 32
• 1.4.2.12 Holder Inequality 32
• 1.4.2.13 Minkowski Inequality 32
3
• 1.4.3 Solution of Linear and Quadratic Inequalities 33
4
• 1.4.3.1 General Remarks 33
• 1.4.3.2 Linear Inequalities 33
• 1.4.3.3 Quadratic Inequalities 33
• 1.4.3.4 General Case for Inequalities of Second Degree 33
3
2
• 1.5 Complex Numbers 34
3
• 1.5.1 Imaginary and Complex Numbers 34
4
• 1.5.1.1 Imaginary Unit 34
• 1.5.1.2 Complex Numbers 34
3
• 1.5.2 Geometric Representation 34
4
• 1.5.2.1 Vector Representation 34
• 1.5.2.2 Equality of Complex Numbers 34
• 1.5.2.3 Trigonometric Form of Complex Numbers 35
• 1.5.2.4 Exponential Form of a Complex Number 35
• 1.5.2.5 Conjugate Complex Numbers 36
3
• 1.5.3 Calculation with Complex Numbers 36
4
• 1.5.3.1 Addition and Subtraction 36
• 1.5.3.2 Multiplication 36
• 1.5.3.3 Division 37
• 1.5.3.4 General Rules for the Basic Operations 37
• 1.5.3.5 Taking Powers of Complex Numbers 37
• 1.5.3.6 Taking of the n-th Root of a Complex Number 38
3
2
• 1.6 Algebraic and Transcendental Equations 38
3
• 1.6.1 Transforming Algebraic Equations to Normal Form 38
4
• 1.6.1.1 Definition 38
• 1.6.1.2 Systems of n Algebraic Equations 38
• 1.6.1.3 Superfluous Roots 39
3
• 1.6.2 Equations of Degree at Most Four 39
4
• 1.6.2.1 Equations of Degree One (Linear Equations) 39
• 1.6.2.2 Equations of Degree Two (Quadratic Equations) 39
• 1.6.2.3 Equations of Degree Three (Cubic Equations) 40
• 1.6.2.4 Equations of Degree Four 42
• 1.6.2.5 Equations of Higher Degree 43
3
• 1.6.3 Equations of Degree n 43
4
• 1.6.3.1 General Properties of Algebraic Equations 43
• 1.6.3.2 Equations with Real Coefficients 44
3
• 1.6.4 Reducing Transcendental Equations to Algebraic Equations 45
4
• 1.6.4.1 Definition 45
• 1.6.4.2 Exponential Equations 45
• 1.6.4.3 Logarithmic Equations 46
• 1.6.4.4 Trigonometric Equations 46
• 1.6.4.5 Equations with Hyperbolic Functions 46
3
2
1
• 2 Functions 47
2
• 2.1 Notion of Functions 47
3
• 2.1.1 Definition of a Function 47
4
• 2.1.1.1 Function 47
• 2.1.1.2 Real Functions 47
• 2.1.1.3 Functions of Several Variables 47
• 2.1.1.4 Complex Functions 47
• 2.1.1.5 Further Functions 47
• 2.1.1.6 Functionals 47
• 2.1.1.7 Functions and Mappings 48
3
• 2.1.2 Methods for Denning a Real Function 48
4
• 2.1.2.1 Defining a Function 48
• 2.1.2.2 Analytic Representation of a Function 48
3
• 2.1.3 Certain Types of Functions 49
4
• 2.1.3.1 Monotone Functions 49
• 2.1.3.2 Bounded Functions 50
• 2.1.3.3 Even Functions 50
• 2.1.3.4 Odd Functions 50
• 2.1.3.5 Representation with Even and Odd Functions 50
• 2.1.3.6 Periodic Functions 50
• 2.1.3.7 Inverse Functions 51
3
• 2.1.4 Limits of Functions 51
4
• 2.1.4.1 Definition of the Limit of a Function 51
• 2.1.4.2 Definition by Limit of Sequences 52
• 2.1.4.3 Cauchy Condition for Convergence 52
• 2.1.4.4 Infinity as a Limit of a Function 52
• 2.1.4.5 Left-Hand and Right-Hand Limit of a Function 52
• 2.1.4.6 Limit of a Function as x Tends to Infinity 53
• 2.1.4.7 Theorems About Limits of Functions 53
• 2.1.4.8 Calculation of Limits 54
• 2.1.4.9 Order of Magnitude of Functions and Landau Order Symbols 55 2.1.5 Continuityof aFunction 57
• 2.1.5.1 Notion of Continuity and Discontinuity 57
• 2.1.5.2 Definition of Continuity 57
• 2.1.5.3 Most Frequent Types of Discontinuities 57
• 2.1.5.4 Continuity and Discontinuity of Elementary Functions 58
• 2.1.5.5 Properties of Continuous Functions 59
3
2
• 2.2 Elementary Functions 60
3
• 2.2.1 Algebraic Functions 60
4
• 2.2.1.1 Polynomials 60
• 2.2.1.2 Rational Functions 61
• 2.2.1.3 Irrational Functions 61
3
• 2.2.2 Transcendental Functions 61
4
• 2.2.2.1 Exponential Functions 61
• 2.2.2.2 Logarithmic Functions 61
• 2.2.2.3 Trigonometric Functions 61
• 2.2.2.4 Inverse Trigonometric Functions 61
• 2.2.2.5 Hyperbolic Functions 62
• 2.2.2.6 Inverse Hyperbolic Functions 62
3
• 2.2.3 Composite Functions 62
2
• 2.3 Polynomials 62
3
• 2.3.1 Linear Function 62
• 2.3.2 Quadratic Polynomial 62
• 2.3.3 Cubic Polynomials 63
• 2.3.4 Polynomials of n-th Degree 63
• 2.3.5 Parabola of n-th Degree 64
2
• 2.4 Rational Functions 64
3
• 2.4.1 Special Fractional Linear Function (Inverse Proportionality) 64
• 2.4.2 Linear Fractional Function 65
• 2.4.3 Curves of Third Degree, Type I 65
• 2.4.4 Curves of Third Degree, Type II 66
• 2.4.5 Curves of Third Degree, Type III 67
• 2.4.6 Reciprocal Powers 68
2
• 2.5 Irrational Functions 69
3
• 2.5.1 Square Root of a Linear Binomial 69
• 2.5.2 Square Root of a Quadratic Polynomial 69
• 2.5.3 Power Function 70
2
• 2.6 Exponential Functions and Logarithmic Functions 71
3
• 2.6.1 Exponential Functions 71
• 2.6.2 Logarithmic Functions 71
• 2.6.3 Error Curve 72
• 2.6.4 Exponential Sum 72
• 2.6.5 Generalized Error Function 73
• 2.6.6 Product of Power and Exponential Functions 74
2
• 2.7 Trigonometric Functions (Functions of Angles) 74
3
• 2.7.1 Basic Notion 74
4
• 2.7.1.1 Definition and Representation 74
• 2.7.1.2 Range and Behavior of the Functions 'i
3
• 2.7.2 Important Formulas for Trigonometric Functions 79
4
• 2.7.2.1 Relations Between the Trigonometric Functions of the Same Angle (Addition Theorems) 79
• 2.7.2.2 Trigonometric Functions of the Sum and Difference of Two Angles 79
• 2.7.2.3 Trigonometric Functions of an Integer Multiple of an Angle 79
• 2.7.2.4 Trigonometric Functions of Half-Angles 80
• 2.7.2.5 Sum and Difference of Two Trigonometric Functions 81
• 2.7.2.6 Products of Trigonometric Functions 81
• 2.7.2.7 Powers of Trigonometric Functions 82
3
• 2.7.3 Description of Oscillations 82
4
• 2.7.3.1 Formulation of the Problem 82
• 2.7.3.2 Superposition of Oscillations 82
• 2.7.3.3 Vector Diagram for Oscillations 83
• 2.7.3.4 Damping of Oscillations 83
3
2
• 2.8 Inverse Trigonometric Functions 84
3
• 2.8.1 Definition of the Inverse Trigonometric Functions 84
• 2.8.2 Reduction to the Principal Value 84
• 2.8.3 Relations Between the Principal Values 85
• 2.8.4 Formulas for Negative Arguments 86
• 2.8.5 Sum and Difference of arcsinx and arcsin y 86
• 2.8.6 Sum and Difference of arccos x and arccos y 86
• 2.8.7 Sum and Difference of arctanx and arctany 86
• 2.8.8 Special Relations for arcsin x, arccos x, arctanx 87
2
• 2.9 Hyperbolic Functions 87
3
• 2.9.1 Definition of Hyperbolic Functions 87
• 2.9.2 Graphical Representation of the Hyperbolic Functions 88
4
• 2.9.2.1 Hyperbolic Sine 88
• 2.9.2.2 Hyperbolic Cosine 88
• 2.9.2.3 Hyperbolic Tangent 88
• 2.9.2.4 Hyperbolic Cotangent 89
3
• 2.9.3 Important Formulas for the Hyperbolic Functions 89
4
• 2.9.3.1 Hyperbolic Functions of One Variable 89
• 2.9.3.2 Expressing a Hyperbolic Function by Another One with the Same Argument 89
• 2.9.3.3 Formulas for Negative Arguments 89
• 2.9.3.4 Hyperbolic Functions of the Sum and Difference of Two Arguments (Addition Theorems) 89
• 2.9.3.5 Hyperbolic Functions of Double Arguments 90
• 2.9.3.6 De Moivre Formula for Hyperbolic Functions 90
• 2.9.3.7 Hyperbolic Functions of Half-Argument 90
• 2.9.3.8 Sum and Difference of Hyperbolic Functions 90
• 2.9.3.9 Relation Between Hyperbolic and Trigonometric Functions with Complex Arguments z 91
3
2
• 2.10 Area Functions 91
3
• 2.10.1 Definitions 91
4
• 2.10.1.1 Area Sine 91
• 2.10.1.2 Area Cosine 91
• 2.10.1.3 Area Tangent 92
• 2.10.1.4 Area Cotangent 92
3
• 2.10.2 Determination of Area Functions Using Natural Logarithm 92
• 2.10.3 Relations Between Different Area Functions 93
• 2.10.4 Sum and Difference of Area Functions 93
• 2.10.5 Formulas for Negative Arguments 93
2
• 2.11 Curves of Order Three (Cubic Curves) 93
3
• 2.11.1 Semicubic Parabola 93
• 2.11.2 Witch of Agnesi 94
• 2.11.3 Cartesian Folium (Folium of Descartes) 94
• 2.11.4 Cissoid 95
• 2.11.5 Strophoide 95
2
• 2.12 Curves of Order Four (Quartics) 96
3
• 2.12.1 Conchoid of Nicomedes 96
• 2.12.2 General Conchoid 96
• 2.12.3 Pascal's Limacon 96
• 2.12.4 Cardioid ' 98
• 2.12.5 Cassinian Curve 98
• 2.12.6 Lemniscate 99
2
• 2.13 Cycloids 100
3
• 2.13.1 Common (Standard) Cycloid 100
• 2.13.2 Prolate and Curtate Cycloids or Trochoids 100
• 2.13.3 Epicycloid 101
• 2.13.4 Hypocycloid and Astroid 102
• 2.13.5 Prolate and Curtate Epicycloid and Hypocycloid 102
2
• 2.14 Spirals 103
3
• 2.14.1 Archimedean Spiral 103
• 2.14.2 Hyperbolic Spiral 104
• 2.14.3 Logarithmic Spiral 104
• 2.14.4 Evolvent of the Circle 104
• 2.14.5 Clothoid 105
2
• 2.15 Various Other Curves 105
3
• 2.15.1 Catenary Curve 105
• 2.15.2 Tractrix 106
2
• 2.16 Determination of Empirical Curves 106
3
• 2.16.1 Procedure 106
4
• 2.16.1.1 Curve-Shape Comparison 106
• 2.16.1.2 Rectification 107
• 2.16.1.3 Determination of Parameters 107
3
• 2.16.2 Useful Empirical Formulas 107
4
• 2.16.2.1 Power Functions 108
• 2.16.2.2 Exponential Functions 108
• 2.16.2.3 Quadratic Polynomial 109
• 2.16.2.4 Rational Linear Function 109
• 2.16.2.5 Square Root of a Quadratic Polynomial 110
• 2.16.2.6 General Error Curve 110
• 2.16.2.7 Curve of Order Three, Type II 110
• 2.16.2.8 Curve of Order Three, Type III Ill
• 2.16.2.9 Curve of Order Three, Type I Ill
• 2.16.2.10 Product of Power and Exponential Functions 112
• 2.16.2.11 Exponential Sum 112
• 2.16.2.12 Numerical Example 112
3
2
• 2.17 Scales and Graph Paper 114
3
• 2.17.1 Scales 114
• 2.17.2 Graph Paper 115
4
• 2.17.2.1 Semilogarithmic Paper 115
• 2.17.2.2 Double Logarithmic Paper 115
• 2.17.2.3 Graph Paper with a Reciprocal Scale 116
• 2.17.2.4 Remark 116
3
2
• 2.18 Functions of Several Variables 117
3
• 2.18.1 Definition and Representation 117
4
• 2.18.1.1 Representation of Functions of Several Variables 117
• 2.18.1.2 Geometric Representation of Functions of Several Variables 117
3
• 2.18.2 Different Domains in the Plane 118
4
• 2.18.2.1 Domain of a Function 118
• 2.18.2.2 Two-Dimensional Domains 118
• 2.18.2.3 Three or Multidimensional Domains 118
• 2.18.2.4 Methods to Determine a Function 119
• 2.18.2.5 Various Ways to Define a Function 120
• 2.18.2.6 Dependence of Functions 121
3
• 2.18.3 Limits 122
4
• 2.18.3.1 Definition 122
• 2.18.3.2 Exact Definition 122
• 2.18.3.3 Generalization for Several Variables 122
• 2.18.3.4 Iterated Limit 122
3
• 2.18.4 Continuity 122
• 2.18.5 Properties of Continuous Functions 123
4
• 2.18.5.1 Theorem on Zeros of Bolzano 123
• 2.18.5.2 Intermediate Value Theorem 123
• 2.18.5.3 Theorem About the Boundedness of a Function 123
• 2.18.5.4 Weierstrass Theorem (About the Existence of Maximum and Minimum) 123
3
2
• 2.19 Nomography 123
3
• 2.19.1 Nomograms 123
• 2.19.2 Net Charts 123
• 2.19.3 Alignment Charts 124
4
• 2.19.3.1 Alignment Charts with Three Straight-Line Scales Through a Point 125
• 2.19.3.2 Alignment Charts with Two Parallel and One Inclined Straight-Line Scales 125
• 2.19.3.3 Alignment Charts with Two Parallel Straight Lines and a Curved Scale 126
3
• 2.19.4 Net Charts for More Than Three Variables 127
2
1
• 3 Geometry 128
2
• 3.1 Plane Geometry 128
3
• 3.1.1 Basic Notation 128
4
• 3.1.1.1 Point. Line, Ray, Segment 128
• 3.1.1.2 Angle 128
• 3.1.1.3 Angle Between Two Intersecting Lines 128
• 3.1.1.4 Pairs of Angles with Intersecting Parallels 129
• 3.1.1.5 Angles Measured in Degrees and in Radians 130
3
• 3.1.2 Geometrical Definition of Circular and Hyperbolic Functions 130
4
• 3.1.2.1 Definition of Circular or Trigonometric Functions 130
• 3.1.2.2 Definitions of the Hyperbolic Functions 131
3
• 3.1.3 Plane Triangles 131
4
• 3.1.3.1 Statements about Plane Triangles 131
• 3.1.3.2 Symmetry 132
3
• 3.1.4 Plane Quadrangles 134
4
• 3.1.4.1 Parallelogram 134
• 3.1.4.2 Rectangle and Square 135
• 3.1.4.3 Rhombus 135
• 3.1.4.4 Trapezoid 135
• 3.1.4.5 General Quadrangle 136
• 3.1.4.6 Inscribed Quadrangle 136
• 3.1.4.7 Circumscribing Quadrangle 136
3
• 3.1.5 Polygons in the Plane 137
4
• 3.1.5.1 General Polygon 137
• 3.1.5.2 Regular Convex Polygons 137
• 3.1.5.3 Some Regular Convex Polygons 138
3
• 3.1.6 The Circle and Related Shapes 138
4
• 3.1.6.1 Circle 138
• 3.1.6.2 Circular Segment and Circular Sector 140
• 3.1.6.3 Annulus 140
3
2
• 3.2 Plane Trigonometry 141
3
• 3.2.1 Triangles 141
4
• 3.2.1.1 Calculations in Right-Angled Triangles in the Plane 141
• 3.2.1.2 Calculations in General Triangles in the Plane 141
3
• 3.2.2 Geodesic Applications 143
4
• 3.2.2.1 Geodetic Coordinates 143
• 3.2.2.2 Angles in Geodesy 145
• 3.2.2.3 Applications in Surveying 14(
3
2
• 3.3 Stereometry 150
3
• 3.3.1 Lines and Planes in Space 150
• 3.3.2 Edge. Corner, Solid Angle 150
• 3.3.3 Polyeder or Polyhedron 151
• 3.3.4 Solids Bounded by Curved Surfaces 154
2
• 3.4 Spherical Trigonometry 158
3
• 3.4.1 Basic Concepts of Geometry on the Sphere 158
4
• 3.4.1.1 Curve. Arc, and Angle on the Sphere 158
• 3.4.1.2 Special Coordinate Systems 160
• 3.4.1.3 Spherical Lune or Biangle 161
• 3.4.1.4 Spherical Triangle 161
• 3.4.1.5 Polar Triangle 162
• 3.4.1.6 Euler Triangles and Non-Euler Triangles 162
• 3.4.1.7 Trihedral Angle 163
3
• 3.4.2 Basic Properties of Spherical Triangles 163
4
• 3.4.2.1 General Statements 163
• 3.4.2.2 Fundamental Formulas and Applications 164
• 3.4.2.3 Further Formulas 166
3
• 3.4.3 Calculation of Spherical Triangles 167
4
• 3.4.3.1 Basic Problems. Accuracy Observations 167
• 3.4.3.2 Right-Angled Spherical Triangles 168
• 3.4.3.3 Spherical Triangles with Oblique Angles 169
• 3.4.3.4 Spherical Curves 172
3
2
• 3.5 Vector Algebra and Analytical Geometry 180
3
• 3.5.1 Vector Algebra 180
4
• 3.5.1.1 Definition of Vectors 180
• 3.5.1.2 Calculation Rules for Vectors 181
• 3.5.1.3 Coordinates of a Vector 182
• 3.5.1.4 Directional Coefficient 183
• 3.5.1.5 Scalar Product and Vector Product 183
• 3.5.1.6 Combination of Vector Products 184
• 3.5.1.7 Vector Equations 187
• 3.5.1.8 Covariant and Contravariant Coordinates of a Vector 187
• 3.5.1.9 Geometric Applications of Vector Algebra 189
3
• 3.5.2 Analytical Geometry of the Plane 189
4
• 3.5.2.1 Basic Concepts, Coordinate Systems in the Plane 189
• 3.5.2.2 Coordinate Transformations 190
• 3.5.2.3 Special Notation in the Plane 191
• 3.5.2.4 Line 194
• 3.5.2.5 Circle 197
• 3.5.2.6 Ellipse 198
• 3.5.2.7 Hyperbola 200
• 3.5.2.8 Parabola 203
• 3.5.2.9 Quadratic Curves (Curves of Second Order or Conic Sections) 205
3
• 3.5.3 Analytical Geometry of Space 207
4
• 3.5.3.1 Basic Concepts, Spatial Coordinate Systems 207
• 3.5.3.2 Transformation of Orthogonal Coordinates 210
• 3.5.3.3 Special Quantities in Space 212
• 3.5.3.4 Line and Plane in Space 214
• 3.5.3.5 Surfaces of Second Order. Equations in Normal Form 220
• 3.5.3.6 Surfaces of Second Order or Quadratic Surfaces. General Theory 223 3.6 Differential Geometry 225
3
• 3.6.1 Plane Curves 225
4
• 3.6.1.1 Ways to Define a Plane Curve 225
• 3.6.1.2 Local Elements of a Curve 225
• 3.6.1.3 Special Points of a Curve 231
• 3.6.1.4 Asymptotes of Curves 234
• 3.6.1.5 General Discussion of a Curve Given by an Equation 235
• 3.6.1.6 Evolutes and Evolvents 236
• 3.6.1.7 Envelope of a Family of Curves 237
3
• 3.6.2 Space Curves 238
4
• 3.6.2.1 Ways to Define a Space Curve 238
• 3.6.2.2 Moving Trihedral 238
• 3.6.2.3 Curvature and Torsion 240
3
• 3.6.3 Surfaces 243
4
• 3.6.3.1 Ways to Define a Surface 243
• 3.6.3.2 Tangent Plane and Surface Normal 244
• 3.6.3.3 Line Elements of a Surface 245
• 3.6.3.4 Curvature of a Surface 247
• 3.6.3.5 Ruled Surfaces and Developable Surfaces 250
• 3.6.3.6 Geodesic Lines on a Surface 250
3
2
1
• 4 Linear Algebra 251
2
• 4.1 Matrices 251
3
• 4.1.1 Notion of Matrix 251
• 4.1.2 Square Matrices 252
• 4.1.3 Vectors 253
• 4.1.4 Arithmetical Operations with Matrices 254
• 4.1.5 Rules of Calculation for Matrices 257
• 4.1.6 Vector and Matrix Norms 258
4
• 4.1.6.1 Vector Norms 258
• 4.1.6.2 Matrix Norms 259
3
2
• 4.2 Determinants 259
3
• 4.2.1 Definitions 259
• 4.2.2 Rules of Calculation for Determinants 260
• 4.2.3 Evaluation of Determinants 261
2
• 4.3 Tensors 262
3
• 4.3.1 Transformation of Coordinate Systems 262
• 4.3.2 Tensors in Cartesian Coordinates 262
• 4.3.3 Tensors with Special Properties 264
4
• 4.3.3.1 Tensors of Rank 2 264
• 4.3.3.2 Invariant Tensors 265
3
• 4.3.4 Tensors in Curvilinear Coordinate Systems 266
4
• 4.3.4.1 Covariant and Contravariant Basis Vectors 266
• 4.3.4.2 Covariant and Contravariant Coordinates of Tensors of Rank 1 266
• 4.3.4.3 Covariant, Contravariant and Mixed Coordinates of Tensors of Rank 2 267
• 4.3.4.4 Rules of Calculation 268
3
• 4.3.5 Pseudotensors 268
4
• 4.3.5.1 Symmetry with Respect to the Origin 269
• 4.3.5.2 Introduction to the Notion of Pseudotensors 270
3
2
• 4.4 Systems of Linear Equations 271
3
• 4.4.1 Linear Systems. Pivoting 271
4
• 4.4.1.1 Linear Systems 271
• 4.4.1.2 Pivoting 271
• 4.4.1.3 Linear Dependence 272
• 4.4.1.4 Calculation of the Inverse of a Matrix 272
3
• 4.4.2 Solution of Systems of Linear Equations 272
4
• 4.4.2.1 Definition and Solvability 272
• 4.4.2.2 Application of Pivoting 274
• 4.4.2.3 Cramer's Rule 275
• 4.4.2.4 Gauss's Algorithm 276
3
• 4.4.3 Overdetermined Linear Equation Systems 277
4
• 4.4.3.1 Overdetermined Linear Systems of Equations and Linear Mean Square Value Problems 277
• 4.4.3.2 Suggestions for Numerical Solutions of Mean Square Value Problems 278
3
2
• 4.5 Eigenvalue Problems for Matrices 278
3
• 4.5.1 General Eigenvalue Problem 278
• 4.5.2 Special Eigenvalue Problem 278
4
• 4.5.2.1 Characteristic Polynomial 278
• 4.5.2.2 Real Symmetric Matrices. Similarity Transformations 280
• 4.5.2.3 Transformation of Principal Axes of Quadratic Forms 281
• 4.5.2.4 Suggestions for the Numerical Calculations of Eigenvalues 283
3
• 4.5.3 Singular Value Decomposition 285
2
1
• 5 Algebra and Discrete Mathematics 286
2
• 5.1 Logic 286
3
• 5.1.1 Propositional Calculus 286
• 5.1.2 Formulas in Predicate Calculus 289
2
• 5.2 Set Theory 290
3
• 5.2.1 Concept of Set, Special Sets 290
• 5.2.2 Operations with Sets 291
• 5.2.3 Relations and Mappings 294
• 5.2.4 Equivalence and Order Relations 296
• 5.2.5 Cardinalityof Sets 298
2
• 5.3 Classical Algebraic Structures 298
3
• 5.3.1 Operations 298
• 5.3.2 Semigroups 299
• 5.3.3 Groups 299
4
• 5.3.3.1 Definition and Basic Properties 299
• 5.3.3.2 Subgroups and Direct Products 300
• 5.3.3.3 Mappings Between Groups 302
3
• 5.3.4 Group Representations 303
4
• 5.3.4.1 Definitions 303
• 5.3.4.2 Particular Representations 303
• 5.3.4.3 Direct Sum of Representations 305
• 5.3.4.4 Direct Product of Representations 305
• 5.3.4.5 Reducible and Irreducible Representations 305
• 5.3.4.6 Schur's Lemma 1 306
• 5.3.4.7 Clebsch Gordan Series 306
• 5.3.4.8 Irreducible Representations of the Symmetric Group S M 306
3
• 5.3.5 Applications of Groups 307
4
• 5.3.5.1 Symmetry Operations, Symmetry Elements 307
• 5.3.5.2 Symmetry Groups or Point Groups 308
• 5.3.5.3 Symmetry Operations with Molecules 308
• 5.3.5.4 Symmetry Groups in Crystallography 310
• 5.3.5.5 Symmetry Groups in Quantum Mechanics 312
• 5.3.5.6 Further Applications of Group Theory in Physics 312
3
• 5.3.6 Rings and Fields 313
4
• 5.3.6.1 Definitions 313
• 5.3.6.2 Subrings, Ideals 313
• 5.3.6.3 Homomorphism, Isomorphism. Homomorphism Theorem 314
• 5.3.6.4 Finite Fields and Shift Registers 314
3
• 5.3.7 Vector Spaces 316
4
• 5.3.7.1 Definition 316
• 5.3.7.2 Linear Dependence 317
• 5.3.7.3 Linear Mappings 317
• 5.3.7.4 Subspaces. Dimension Formula 317
• 5.3.7.5 Euclidean Vector Spaces. Euclidean Norm 318
• 5.3.7.6 Linear Operators in Vector Spaces 319
3
2
• 5.4 Elementary Number Theory 320
3
• 5.4.1 Divisibility 320
4
• 5.4.1.1 Divisibility and Elementary Divisibility Rules 320
• 5.4.1.2 Prime Numbers 320
• 5.4.1.3 Criteria for Divisibility 322
• 5.4.1.4 Greatest Common Divisor and Least Common Multiple 323
• 5.4.1.5 Fibonacci Numbers 325
3
• 5.4.2 Linear Diophantine Equations 325
• 5.4.3 Congruences and Residue Classes 327
• 5.4.4 Theorems of Fermat, Euler. and Wilson 331
• 5.4.5 Codes 331
2
• 5.5 Cryptology 334
3
• 5.5.1 Problem of Cryptology 334
• 5.5.2 Cryptosystems 334
• 5.5.3 Mathematical Foundation 334
• 5.5.4 Security of Cryptosystems 335
4
• 5.5.4.1 Methods of Conventional Cryptography 335
• 5.5.4.2 Linear Substitution Ciphers 336
• 5.5.4.3 Vigenere Cipher 336
• 5.5.4.4 Matrix Substitution 336
3
• 5.5.5 Methods of Classical Cryptanalysis 337
4
• 5.5.5.1 Statistical Analysis 337
• 5.5.5.2 Kasiski Friedman Test 337
3
• 5.5.6 One-Time Pad 338
• 5.5.7 Public Key Methods 338
4
• 5.5.7.1 Diffie Hellman Key Exchange 338
• 5.5.7.2 One-Way Function 339
• 5.5.7.3 RSA Method 339
3
• 5.5.8 AES Algorithm (Advanced Encryption Standard) 339
• 5.5.9 IDEA Algorithm (International Data Encryption Algorithm) 340
2
• 5.6 Universal Algebra 340
3
• 5.6.1 Definition 340
• 5.6.2 Congruence Relations. Factor Algebras 340
• 5.6.3 Homomorphism 341
• 5.6.4 Homomorphism Theorem 341
• 5.6.5 Varieties 341
• 5.6.6 Term Algebras. Free Algebras 341
2
• 5.7 Boolean Algebras and Switch Algebra 342
3
• 5.7.1 Definition 342
• 5.7.2 Duality Principle 343
• 5.7.3 Finite Boolean Algebras 343
• 5.7.4 Boolean Algebras as Orderings 343
• 5.7.5 Boolean Functions. Boolean Expressions 344
• 5.7.6 Normal Forms 345
• 5.7.7 Switch Algebra 346
2
• 5.8 Algorithms of Graph Theory 348
3
• 5.8.1 Basic Notions and Notation 348
• 5.8.2 Traverse of Undirected Graphs 351
4
• 5.8.2.1 Edge Sequences or Paths 351
• 5.8.2.2 Euler Trails 352
• 5.8.2.3 Hamiltonian Cycles 353
3
• 5.8.3 Trees and Spanning Trees 354
4
• 5.8.3.1 Trees 354
• 5.8.3.2 Spanning Trees 355
3
• 5.8.4 Matchings 356
• 5.8.5 Planar Graphs 357
• 5.8.6 Paths in Directed Graphs 357
• 5.8.7 Transport Networks 358
2
• 5.9 Fuzzy Logic 360
3
• 5.9.1 Basic Notions of Fuzzy Logic 360
4
• 5.9.1.1 Interpretation of Fuzzy Sets 360
• 5.9.1.2 Membership Functions on the Real Line 361
• 5.9.1.3 Fuzzy Sets 363
3
• 5.9.2 Aggregation of Fuzzy Sets 365
4
• 5.9.2.1 Concepts for Aggregation of Fuzzy Sets 365
• 5.9.2.2 Practical Aggregator Operations of Fuzzy Sets 366
• 5.9.2.3 Compensatory Operators 368
• 5.9.2.4 Extension Principle 368
• 5.9.2.5 Fuzzy Complement 368
3
• 5.9.3 Fuzzy-Valued Relations 369
4
• 5.9.3.1 Fuzzy Relations 369
• 5.9.3.2 Fuzzy Product Relation R o S 371
3
• 5.9.4 Fuzzy Inference (Approximate Reasoning) 372
• 5.9.5 Defuzzification Methods 373
• 5.9.6 Knowledge-Based Fuzzy Systems 374
4
• 5.9.6.1 Method of Mamdani 374
• 5.9.6.2 Method of Sugeno 375
• 5.9.6.3 Cognitive Systems 375
• 5.9.6.4 Knowledge-Based Interpolation Systems 377
3
2
1
• 6 Differentiation 379
2
• 6.1 Differentiation of Functions of One Variable 379
3
• 6.1.1 Differential Quotient 379
• 6.1.2 Rules of Differentiation for Functions of One Variable 380
4
• 6.1.2.1 Derivatives of the Elementary Functions 380
• 6.1.2.2 Basic Rules of Differentiation 380
3
• 6.1.3 Derivatives of Higher Order 385
4
• 6.1.3.1 Definition of Derivatives of Higher Order 385
• 6.1.3.2 Derivatives of Higher Order of some Elementary Functions 385
• 6.1.3.3 Leibniz's Formula 385
• 6.1.3.4 Higher Derivatives of Functions Given in Parametric Form 387
• 6.1.3.5 Derivatives of Higher Order of the Inverse Function 387
3
• 6.1.4 Fundamental Theorems of Differential Calculus 388
4
• 6.1.4.1 Monotonicity 388
• 6.1.4.2 Fermat's Theorem 388
• 6.1.4.3 Rolle's Theorem 388
• 6.1.4.4 Mean Value Theorem of Differential Calculus 389
• 6.1.4.5 Taylor's Theorem of Functions of One Variable 389
• 6.1.4.6 Generalized Mean Value Theorem of Differential Calculus (Cauchy's Theorem) 390
3
• 6.1.5 Determination of the Extreme Values and Inflection Points 390
4
• 6.1.5.1 Maxima and Minima 390
• 6.1.5.2 Necessary Conditions for the Existence of a Relative Extreme Value 390
• 6.1.5.3 Relative Extreme Values of a Differentiable, Explicit Function 391
• 6.1.5.4 Determination of Absolute Extrema 392
• 6.1.5.5 Determination of the Extrema of Implicit Functions 392
3
2
• 6.2 Differentiation of Functions of Several Variables 392
3
• 6.2.1 Partial Derivatives 392
4
• 6.2.1.1 Partial Derivativeofa Function 392
• 6.2.1.2 Geometrical Meaning for Functions of Two Variables 393
• 6.2.1.3 Differentials of x and f(x) 393
• 6.2.1.4 Basic Properties of the Differential 394
• 6.2.1.5 Partial Differential 394
3
• 6.2.2 Total Differential and Differentials of Higher Order 394
4
• 6.2.2.1 Notion of Total Differential of a Function of Several Variables (Complete Differential) 394
• 6.2.2.2 Derivatives and Differentials of Higher Order 395
• 6.2.2.3 Taylor's Theorem for Functions of Several Variables 396
3
• 6.2.3 Rules of Differentiation for Functions of Several Variables 397
4
• 6.2.3.1 Differentiation of Composite Functions 397
• 6.2.3.2 Differentiation of Implicit Functions 398
3
• 6.2.4 Substitution of Variables in Differential Expressions and Coordinate Transformations 399
4
• 6.2.4.1 Function of One Variable 399
• 6.2.4.2 Function of Two Variables 400
3
• 6.2.5 Extreme Values of Functions of Several Variables 401
4
• 6.2.5.1 Definition 401
• 6.2.5.2 Geometric Representation 401
• 6.2.5.3 Determination of Extreme Values of Functions of Two Variables 402
• 6.2.5.4 Determination of the Extreme Values of a Function of n Variables 402
• 6.2.5.5 Solution of Approximation Problems 403
• 6.2.5.6 Extreme Value Problem with Side Conditions 403
3
2
1
• 7 Infinite Series 404
2
• 7.1 Sequences of Numbers 404
3
• 7.1.1 Properties of Sequences of Numbers 404
4
• 7.1.1.1 Definition of Sequence of Numbers 404
• 7.1.1.2 Monotone Sequences of Numbers 404
• 7.1.1.3 Bounded Sequences 404
3
• 7.1.2 Limits of Sequences of Numbers 405
2
• 7.2 Number Series 406
3
• 7.2.1 General Convergence Theorems 406
4
• 7.2.1.1 Convergence and Divergence of Infinite Series 406
• 7.2.1.2 General Theorems about the Convergence of Series 406
3
• 7.2.2 Convergence Criteria for Series with Positive Terms 407
4
• 7.2.2.1 Comparison Criterion 407
• 7.2.2.2 D'Alembert's Ratio Test 407
• 7.2.2.3 Root Test of Cauchy 408
• 7.2.2.4 Integral Test of Cauchy 408
3
• 7.2.3 Absolute and Conditional Convergence 409
4
• 7.2.3.1 Definition 409
• 7.2.3.2 Properties of Absolutely Convergent Series 409
• 7.2.3.3 Alternating Series 410
3
• 7.2.4 Some Special Series 410
4
• 7.2.4.1 The Values of Some Important Number Series 410
• 7.2.4.2 Bernoulli and Euler Numbers 412
3
• 7.2.5 Estimation of the Remainder 413
4
• 7.2.5.1 Estimation with Majorant 413
• 7.2.5.2 Alternating Convergent Series 414
• 7.2.5.3 Special Series 414
3
2
• 7.3 Function Series 414
3
• 7.3.1 Definitions 414
• 7.3.2 Uniform Convergence 414
4
• 7.3.2.1 Definition. Weierstrass Theorem 414
• 7.3.2.2 Properties of Uniformly Convergent Series 415
3
• 7.3.3 Power series 416
4
• 7.3.3.1 Definition. Convergence 416
• 7.3.3.2 Calculations with Power Series 416
• 7.3.3.3 Taylor Series Expansion, Maclaurin Series 417
3
• 7.3.4 Approximation Formulas 418
• 7.3.5 Asymptotic Power Series 419
4
• 7.3.5.1 Asymptotic Behavior 419
• 7.3.5.2 Asymptotic Power Series 420
3
2
• 7.4 Fourier Series 420
3
• 7.4.1 Trigonometric Sum and Fourier Series 420
4
• 7.4.1.1 Basic Notions 420
• 7.4.1.2 Most Important Properties of the Fourier Series 421
3
• 7.4.2 Determination of Coefficients for Symmetric Functions 422
4
• 7.4.2.1 Different Kinds of Symmetries 422
• 7.4.2.2 Forms of the Expansion into a Fourier Series 423
3
• 7.4.3 Determination of the Fourier Coefficients with Numerical Methods 424
• 7.4.4 Fourier Series and Fourier Integrals 424
• 7.4.5 Remarks on the Table of Some Fourier Expansions 425
2
1
• 8 Integral Calculus 427
2
• 8.1 Indefinite Integrals 427
3
• 8.1.1 Primitive Function or Antiderivative 427
4
• 8.1.1.1 Indefinite Integrals 428
• 8.1.1.2 Integrals of Elementary Functions 428
3
• 8.1.2 Rules of Integration 429
• 8.1.3 Integration of Rational Functions 432
4
• 8.1.3.1 Integrals of Integer Rational Functions (Polynomials) 432
• 8.1.3.2 Integrals of Fractional Rational Functions 432
• 8.1.3.3 Four Cases of Partial Fraction Decomposition 432
3
• 8.1.4 Integration of Irrational Functions 435
4
• 8.1.4.1 Substitution to Reduce to Integration of Rational Functions 435
• 8.1.4.2 Integration of Binomial Integrands 436
• 8.1.4.3 Elliptic Integrals 437
3
• 8.1.5 Integration of Trigonometric Functions 438
4
• 8.1.5.1 Substitution 438
• 8.1.5.2 Simplified Methods 438
3
• 8.1.6 Integration of Further Transcendental Functions 439
4
• 8.1.6.1 Integrals with Exponential Functions 439
• 8.1.6.2 Integrals with Hyperbolic Functions 440
• 8.1.6.3 Application of Integration by Parts 440
• 8.1.6.4 Integrals of Transcendental Functions 440
3
2
• 8.2 Definite Integrals 440
3
• 8.2.1 Basic Notions. Rules and Theorems 440
4
• 8.2.1.1 Definition and Existence of the Definite Integral 440
• 8.2.1.2 Properties of Definite Integrals 441
• 8.2.1.3 Further Theorems about the Limits of Integration 443
• 8.2.1.4 Evaluation of the Definite Integral 445
3
• 8.2.2 Application of Definite Integrals 447
4
• 8.2.2.1 General Principles for Application of the Definite Integral 447
• 8.2.2.2 Applications in Geometry 448
• 8.2.2.3 Applications in Mechanics and Physics 451
3
• 8.2.3 Improper Integrals, Stieltjes and Lebesgue Integrals 453
4
• 8.2.3.1 Generalization of the Notion of the Integral 453
• 8.2.3.2 Integrals with Infinite Integration Limits 454
• 8.2.3.3 Integrals with Unbounded Integrand 456
3
• 8.2.4 Parametric Integrals 459
4
• 8.2.4.1 Definition of Parametric Integrals 459
• 8.2.4.2 Differentiation Under the Symbol of Integration 459
• 8.2.4.3 Integration Under the Symbol of Integration 459
3
• 8.2.5 Integration by Series Expansion. Special Non-Elementary Functions 460
2
• 8.3 Line Integrals 462
3
• 8.3.1 Line Integrals of the First Type 463
4
• 8.3.1.1 Definitions 463
• 8.3.1.2 Existence Theorem 463
• 8.3.1.3 Evaluation of the Line Integral of the First Type 463
• 8.3.1.4 Application of the Line Integral of the First Type 464
3
• 8.3.2 Line Integrals of the Second Type 464
4
• 8.3.2.1 Definitions 464
• 8.3.2.2 Existence Theorem 466
• 8.3.2.3 Calculation of the Line Integral of the Second Type 466
3
• 8.3.3 Line Integrals of General Type 467
4
• 8.3.3.1 Definition 467
• 8.3.3.2 Properties of the Line Integral of General Type 467
• 8.3.3.3 Integral Along a Closed Curve 468
3
• 8.3.4 Independence of the Line Integral of the Path of Integration 468
4
• 8.3.4.1 Two-Dimensional Case 468
• 8.3.4.2 Existence of a Primitive Function 469
• 8.3.4.3 Three-Dimensional Case 469
• 8.3.4.4 Determination of the Primitive Function 469
• 8.3.4.5 Zero-Valued Integral Along a Closed Curve 470
3
2
• 8.4 Multiple Integrals 471
3
• 8.4.1 Double Integrals 471
4
• 8.4.1.1 Notion of the Double Integral 471
• 8.4.1.2 Evaluation of the Double Integral 472
• 8.4.1.3 Applications of the Double Integral 474
3
• 8.4.2 Triple Integrals 476
4
• 8.4.2.1 Notion of the Triple Integral 476
• 8.4.2.2 Evaluation of the Triple Integral 476
• 8.4.2.3 Applications of the Triple Integral 479
3
2
• 8.5 Surface Integrals 479
3
• 8.5.1 Surface Integral of the First Type 479
4
• 8.5.1.1 Notion of the Surface Integral of the First Type 480
• 8.5.1.2 Evaluation of the Surface Integral of the First Type 481
• 8.5.1.3 Applications of the Surface Integral of the First Type 482
3
• 8.5.2 Surface Integral of the Second Type 483
4
• 8.5.2.1 Notion of the Surface Integral of the Second Type 483
• 8.5.2.2 Evaluation of Surface Integrals of the Second Type 484
3
• 8.5.3 Surface Integral in General Form 485
4
• 8.5.3.1 Notion of the Surface Integral in General Form 485
• 8.5.3.2 Properties of the Surface Integrals 485
• 8.5.3.3 An Application of the Surface Integral 486
3
2
1
• 9 Differential Equations 487
2
• 9.1 Ordinary Differential Equations 487
3
• 9.1.1 First-Order Differential Equations 488
4
• 9.1.1.1 Existence Theorems. Direction Field 488
• 9.1.1.2 Important Solution Methods 489
• 9.1.1.3 Implicit Differential Equations 492
• 9.1.1.4 Singular Integrals and Singular Points 493
• 9.1.1.5 Approximation Methods for Solution of First-Order Differential Equations 496
3
• 9.1.2 Differential Equations of Higher Order and Systems of Differential Equations 497
4
• 9.1.2.1 Basic Results 497
• 9.1.2.2 Lowering the Order 499 I
• 9.1.2.3 Linear n-th Order Differential Equations 500
• 9.1.2.4 Solution of Linear Differential Equations with Constant Coefficients 502
• 9.1.2.5 Systems of Linear Differential Equations with Constant Coefficients 505
• 9.1.2.6 Linear Second-Order Differential Equations 507
3
• 9.1.3 Boundary Value Problems 514
4
• 9.1.3.1 Problem Formulation 514
• 9.1.3.2 Fundamental Properties of Eigenfunctions and Eigenvalues 515
• 9.1.3.3 Expansion in Eigenfunctions 516
• 9.1.3.4 Singular Cases 516
3
2
• 9.2 Partial Differential Equations 517
3
• 9.2.1 First-Order Partial Differential Equations 517
4
• 9.2.1.1 Linear First-Order Partial Differential Equations 517
• 9.2.1.2 Non-Linear First-Order Partial Differential Equations 519
3
• 9.2.2 Linear Second-Order Partial Differential Equations 522
4
• 9.2.2.1 Classification and Properties of Second-Order Differential Equations with Two Independent Variables 522
• 9.2.2.2 Classification and Properties of Linear Second-Order Differential Equations with more than two Independent Variables 523
• 9.2.2.3 Integration Methods for Linear Second-Order Partial Differential Equations 524
3
• 9.2.3 Some further Partial Differential Equations From Natural Sciences and Engineering 534
4
• 9.2.3.1 Formulation of the Problem and the Boundary Conditions 534
• 9.2.3.2 Wave Equation 536
• 9.2.3.3 Heat Conduction and Diffusion Equation for Homogeneous Media 537
• 9.2.3.4 Potential Equation 538
• 9.2.3.5 Schrodinger's Equation 538
3
• 9.2.4 Non-Linear Partial Differential Equations: Solitons, Periodic Patterns and Chaos 546
4
• 9.2.4.1 Formulation of the Physical-Mathematical Problem 546
• 9.2.4.2 Korteweg de Vries Equation (KdV) 548
• 9.2.4.3 Non-Linear Schrodinger Equation (NLS) 549
• 9.2.4.4 Sine Gordon Equation (SG) 549
• 9.2.4.5 Further Non-linear Evolution Equations with Soliton Solutions 551
3
2
1
• 10 Calculus of Variations 552
2
• 10.1 Defining the Problem 552
• 10.2 Historical Problems 553
3
• 10.2.1 Isoperimetric Problem 553
• 10.2.2 Brachistochrone Problem 553
2
• 10.3 Variational Problems of One Variable 553
3
• 10.3.1 Simple Variational Problems and Extremal Curves 553
• 10.3.2 Euler Differential Equation of the Variational Calculus 554
• 10.3.3 Variational Problems with Side Conditions 555
• 10.3.4 Variational Problems with Higher-Order Derivatives 556
• 10.3.5 Variational Problem with Several Unknown Functions 557
• 10.3.6 Variational Problems using Parametric Representation 557
2
• 10.4 Variational Problems with Functions of Several Variables 558
3
• 10.4.1 Simple Variational Problem 558
• 10.4.2 More General Variational Problems 560
2
• 10.5 Numerical Solution of Variational Problems 560
• 10.6 Supplementary Problems 561
3
• 10.6.1 First and Second Variation 561
• 10.6.2 Application in Physics 562
2
1
• 11 Linear Integral Equations 563
2
• 11.1 Introduction and Classification 563
• 11.2 Fredholm Integral Equations of the Second Kind 564
3
• 11.2.1 Integral Equations with Degenerate Kernel 564
• 11.2.2 Successive Approximation Method, Neumann Series 567
• 11.2.3 Fredholm Solution Method. Fredholm Theorems 569
4
• 11.2.3.1 Fredholm Solution Method 569
• 11.2.3.2 Fredholm Theorems 571
3
• 11.2.4 Numerical Methods for Fredholm Integral Equations of the Second Kind 572
4
• 11.2.4.1 Approximation of the Integral 572
• 11.2.4.2 Kernel Approximation 574
• 11.2.4.3 Collocation Method 576
3
2
• 11.3 Fredholm Integral Equations of the First Kind 577
3
• 11.3.1 Integral Equations with Degenerate Kernels 577
• 11.3.2 Analytic Basis 578
• 11.3.3 Reduction of an Integral Equation into a Linear System of Equations 580
• 11.3.4 Solution of the Homogeneous Integral Equation of the First Kind 581
• 11.3.5 Construction of Two Special Orthonormal Systems for a Given Kernel 582
• 11.3.6 Iteration Method 584
2
• 11.4 Volterra Integral Equations 585
3
• 11.4.1 Theoretical Foundations 585
• 11.4.2 Solution by Differentiation 586
• 11.4.3 Solution of the Volterra Integral Equation of the Second Kind by Neumann Series 587
• 11.4.4 Convolution Type Volterra Integral Equations 587
• 11.4.5 Numerical Methods for Volterra Integral Equations of the Second Kind 588
2
• 11.5 Singular Integral Equations 590
3
• 11.5.1 Abel Integral Equation 590
• 11.5.2 Singular Integral Equation with Cauchy Kernel 591
4
• 11.5.2.1 Formulation of the Problem 591
• 11.5.2.2 Existence of a Solution 592
• 11.5.2.3 Properties of Cauchy Type Integrals 592
• 11.5.2.4 The Hilbert Boundary Value Problem 593
• 11.5.2.5 Solution of the Hilbert Boundary Value Problem (in short: Hilbert Problem) 593
• 11.5.2.6 Solution of the Characteristic Integral Equation 594
3
2
1
• 12 Functional Analysis 596
2
• 12.1 Vector Spaces 596
3
• 12.1.1 Notion of a Vector Space 596
• 12.1.2 Linear and Affine Linear Subsets 597
• 12.1.3 Linearly Independent Elements 598
• 12.1.4 Convex Subsets and the Convex Hull 599
4
• 12.1.4.1 Convex Sets 599
• 12.1.4.2 Cones 599
3
• 12.1.5 Linear Operators and Functionals 600
4
• 12.1.5.1 Mappings 600
• 12.1.5.2 Homomorphism and Endomorphism 600
• 12.1.5.3 Isomorphic Vector Spaces 601
3
• 12.1.6 Complexification of Real Vector Spaces 601
• 12.1.7 Ordered Vector Spaces 601
4
• 12.1.7.1 Cone and Partial Ordering 601
• 12.1.7.2 Order Bounded Sets 602
• 12.1.7.3 Positive Operators 602
• 12.1.7.4 Vector Lattices 602
3
2
• 12.2 Metric Spaces 604
3
• 12.2.1 Notion of a Metric Space 604
4
• 12.2.1.1 Balls, Neighborhoods and Open Sets 605
• 12.2.1.2 Convergence of Sequences in Metric Spaces 606
• 12.2.1.3 Closed Sets and Closure 606
• 12.2.1.4 Dense Subsets and Separable Metric Spaces 607
3
• 12.2.2 CompleteMetricSpaces 607
4
• 12.2.2.1 Cauchy Sequences 607
• 12.2.2.2 CompleteMetricSpaces 608
• 12.2.2.3 Some Fundamental Theorems in Complete Metric Spaces 608
• 12.2.2.4 Some Applications of the Contraction Mapping Principle 608
• 12.2.2.5 Completion of a Metric Space 610
3
• 12.2.3 Continuous Operators 610
2
• 12.3 NormedSpaces 611
3
• 12.3.1 Notion of a Normed Space 611
4
• 12.3.1.1 Axioms of a Normed Space 611
• 12.3.1.2 Some Properties of Normed Spaces 612
3
• 12.3.2 Banach Spaces 612
4
• 12.3.2.1 Series in Normed Spaces 612
• 12.3.2.2 Examples of Banach Spaces 612
• 12.3.2.3 Sobolev Spaces 613
3
• 12.3.3 Ordered Normed Spaces 613
• 12.3.4 Normed Algebras 614
2
• 12.4 HilbertSpaces 615
3
• 12.4.1 Notion of a Hilbert Space 615
4
• 12.4.1.1 Scalar Product 615
• 12.4.1.2 Unitary Spaces and Some of their Properties 615
• 12.4.1.3 Hilbert Space 615
3
• 12.4.2 Orthogonality 616
4
• 12.4.2.1 Properties of Orthogonality 616
• 12.4.2.2 Orthogonal Systems 616
3
• 12.4.3 Fourier Series in Hilbert Spaces 617
4
• 12.4.3.1 Best Approximation 617
• 12.4.3.2 Parseval Equation, Riesz Fischer Theorem 618
3
• 12.4.4 Existence of a Basis, Isomorphic Hilbert Spaces 618
2
• 12.5 Continuous Linear Operators and Functionals 619
3
• 12.5.1 Boundedness, Norm and Continuity of Linear Operators 619
4
• 12.5.1.1 Boundedness and the Norm of Linear Operators 619
• 12.5.1.2 The Space of Linear Continuous Operators 619
• 12.5.1.3 Convergence of Operator Sequences 620
3
• 12.5.2 Linear Continuous Operators in Banach Spaces 620
• 12.5.3 Elements of the Spectral Theory of Linear Operators 622
4
• 12.5.3.1 Resolvent Set and the Resolvent of an Operator 622
• 12.5.3.2 Spectrum of an Operator 622
3
• 12.5.4 Continuous Linear Functionals 623
4
• 12.5.4.1 Definition 623
• 12.5.4.2 Continuous Linear Functionals in Hilbert Spaces. Riesz Representation Theorem 624
• 12.5.4.3 Continuous Linear Functionals in L p 624
3
• 12.5.5 Extension of a Linear Functional 624
• 12.5.6 Separation of Convex Sets 625
• 12.5.7 Second Adjoint Space and Reflexive Spaces 626
2
• 12.6 Adjoint Operators in Normed Spaces 626
3
• 12.6.1 Adjoint of a Bounded Operator 626
• 12.6.2 Adjoint Operator of an Unbounded Operator 627
• 12.6.3 Self-Adjoint Operators 627
4
• 12.6.3.1 Positive Definite Operators 628
• 12.6.3.2 Projectors in a Hilbert Space 628
3
2
• 12.7 Compact Sets and Compact Operators 628
3
• 12.7.1 Compact Subsets of a Normed Space 628
• 12.7.2 Compact Operators 628
4
• 12.7.2.1 Definition of Compact Operator 628
• 12.7.2.2 Properties of Linear Compact Operators 629
• 12.7.2.3 Weak Convergence of Elements 629
3
• 12.7.3 Fredholm Alternative 629
• 12.7.4 Compact Operators in Hilbert Space 630
• 12.7.5 Compact Self-Adjoint Operators 630
2
• 12.8 Non-Linear Operators 631
3
• 12.8.1 Examples of Non-Linear Operators 631
• 12.8.2 Differentiability of Non-Linear Operators 632
• 12.8.3 Newton's Method 632
• 12.8.4 Schauder's Fixed-Point Theorem 633
• 12.8.5 Leray Schauder Theory 633
• 12.8.6 Positive Non-Linear Operators 633
• 12.8.7 Monotone Operators in Banach Spaces 634
2
• 12.9 Measure and Lebesgue Integral 635
3
• 12.9.1 Sigma Algebra and Measures 635
• 12.9.2 Measurable Functions 636
4
• 12.9.2.1 Measurable Function 636
• 12.9.2.2 Properties of the Class of Measurable Functions 636
3
• 12.9.3 Integration 637
4
• 12.9.3.1 Definition of the Integral 637
• 12.9.3.2 Some Properties of the Integral 637
• 12.9.3.3 Convergence Theorems 638
3
• 12.9.4 p Spaces 639
• 12.9.5 Distributions 639
4
• 12.9.5.1 Formula of Partial Integration 639
• 12.9.5.2 Generalized Derivative 640
• 12.9.5.3 Distributions 640
• 12.9.5.4 Derivative of a Distribution 641
3
2
1
• 13 Vector Analysis and Vector Fields 642
2
• 13.1 Basic Notions of the Theory of Vector Fields 642
3
• 13.1.1 Vector Functions of a Scalar Variable 642
4
• 13.1.1.1 Definitions 642
• 13.1.1.2 Derivativeofa Vector Function 642
• 13.1.1.3 Rules of Differentiation for Vectors 642
• 13.1.1.4 Taylor Expansion for Vector Functions 643
3
• 13.1.2 Scalar Fields 643
4
• 13.1.2.1 Scalar Field or Scalar Point Function 643
• 13.1.2.2 Important Special Cases of Scalar Fields 643
• 13.1.2.3 Coordinate Definition of a Field 644
• 13.1.2.4 Level Surfaces and Level Lines of a Field 644
3
• 13.1.3 Vector Fields 645
4
• 13.1.3.1 Vector Field or Vector Point Function 645
• 13.1.3.2 Important Cases of Vector Fields 645
• 13.1.3.3 Coordinate Representation of Vector Fields 646
• 13.1.3.4 Transformation of Coordinate Systems 647
• 13.1.3.5 Vector Lines 648
3
2
• 13.2 Differential Operators of Space 649
3
• 13.2.1 Directional and Space Derivatives 649
4
• 13.2.1.1 Directional Derivative of a Scalar Field 649
• 13.2.1.2 Directional Derivative of a Vector Field 650
• 13.2.1.3 Volume Derivative 650
3
• 13.2.2 Gradient of a Scalar Field 650
4
• 13.2.2.1 Definition of the Gradient 651
• 13.2.2.2 Gradient and Volume Derivative 651
• 13.2.2.3 Gradient and Directional Derivative 651
• 13.2.2.4 Further Properties of the Gradient 651
• 13.2.2.5 Gradient of the Scalar Field in Different Coordinates 651
• 13.2.2.6 Rules of Calculations 652
3
• 13.2.3 Vector Gradient 652
• 13.2.4 Divergence of Vector Fields 653
4
• 13.2.4.1 Definition of Divergence 653
• 13.2.4.2 Divergence in Different Coordinates 653
• 13.2.4.3 Rules for Evaluation of the Divergence 653
• 13.2.4.4 Divergence of a Central Field 654
3
• 13.2.5 Rotation of Vector Fields 654
4
• 13.2.5.1 Definitions of the Rotation 654
• 13.2.5.2 Rotation in Different Coordinates 655
• 13.2.5.3 Rules for Evaluating the Rotation 655
• 13.2.5.4 Rotation of a Potential Field 656
3
• 13.2.6 Nabla Operator, Laplace Operator 656
4
• 13.2.6.1 Nabla Operator 656
• 13.2.6.2 Rules for Calculations with the Nabla Operator 656
• 13.2.6.3 Vector Gradient 657
• 13.2.6.4 Nabla Operator Applied Twice 657
• 13.2.6.5 Laplace Operator 657
3
• 13.2.7 Review of Spatial Differential Operations 658
4
• 13.2.7.1 Fundamental Relations and Results (see Table 13.2) 658
• 13.2.7.2 Rules of Calculation for Spatial Differential Operators 658
• 13.2.7.3 Expressions of Vector Analysis in Cartesian. Cylindrical, and Spherical Coordinates 659
3
2
• 13.3 Integration in Vector Fields 660
3
• 13.3.1 Line Integral and Potential in Vector Fields 660
4
• 13.3.1.1 Line Integral in Vector Fields 660
• 13.3.1.2 Interpretation of the Line Integral in Mechanics 661
• 13.3.1.3 Properties of the Line Integral 661
• 13.3.1.4 Line Integral in Cartesian Coordinates 661
• 13.3.1.5 Integral Along a Closed Curve in a Vector Field 662
• 13.3.1.6 Conservative or Potential Field 662
3
• 13.3.2 Surface Integrals 663
4
• 13.3.2.1 Vector of a Plane Sheet 663
• 13.3.2.2 Evaluation of the Surface Integral 663
• 13.3.2.3 Surface Integrals and Flow of Fields 664
• 13.3.2.4 Surface Integrals in Cartesian Coordinates as Surface Integral of Second Type 664
3
• 13.3.3 Integral Theorems 665
4
• 13.3.3.1 Integral Theorem and Integral Formula of Gauss 665
• 13.3.3.2 Integral Theorem of Stokes 666
• 13.3.3.3 Integral Theorems of Green 666
3
2
• 13.4 Evaluation of Fields 667
3
• 13.4.1 Pure Source Fields 667
• 13.4.2 Pure Rotation Field or Zero-Divergence Field 668
• 13.4.3 Vector Fields with Point-Like Sources 668
4
• 13.4.3.1 Coulomb Field of a Point-Like Charge 668
• 13.4.3.2 Gravitational Field of a Point Mass 669
3
• 13.4.4 Superposition of Fields 669
4
• 13.4.4.1 Discrete Source Distribution 669
• 13.4.4.2 Continuous Source Distribution 669
• 13.4.4.3 Conclusion 669
3
2
• 13.5 Differential Equations of Vector Field Theory 669
3
• 13.5.1 Laplace Differential Equation 669
• 13.5.2 Poisson Differential Equation 670
2
1
• 14 Function Theory 671
2
• 14.1 Functions of Complex Variables 671
3
• 14.1.1 Continuity, Differentiability 671
4
• 14.1.1.1 Definition of a Complex Function 671
• 14.1.1.2 Limit of a Complex Function 671
• 14.1.1.3 Continuous Complex Functions 671
• 14.1.1.4 Differentiability of a Complex Function 671
3
• 14.1.2 Analytic Functions 672
4
• 14.1.2.1 Definition of Analytic Functions 672
• 14.1.2.2 Examples of Analytic Functions 672
• 14.1.2.3 Properties of Analytic Functions 672
• 14.1.2.4 Singular Points 673
3
• 14.1.3 Conformal Mapping 674
4
• 14.1.3.1 Notion and Properties of Conformal Mappings 674
• 14.1.3.2 Simplest Conformal Mappings 675
• 14.1.3.3 The Schwarz Reflection Principle 681
• 14.1.3.4 Complex Potential 681
• 14.1.3.5 Superposition Principle 683
• 14.1.3.6 Arbitrary Mappings of the Complex Plane 684
3
2
• 14.2 Integration in the Complex Plane 685
3
• 14.2.1 Definite and Indefinite Integral 685
4
• 14.2.1.1 Definition of the Integral in the Complex Plane 685
• 14.2.1.2 Properties and Evaluation of Complex Integrals 686
3
• 14.2.2 Cauchy Integral Theorem 688
4
• 14.2.2.1 Cauchy Integral Theorem for Simply Connected Domains 688
• 14.2.2.2 Cauchy Integral Theorem for Multiply Connected Domains 688
3
• 14.2.3 Cauchy Integral Formulas 689
4
• 14.2.3.1 Analytic Function on the Interior of a Domain 689
• 14.2.3.2 Analytic Function on the Exterior of a Domain 689
3
2
• 14.3 Power Series Expansion of Analytic Functions 689
3
• 14.3.1 Convergence of Series with Complex Terms 689
4
• 14.3.1.1 Convergence of a Number Sequence with Complex Terms 689
• 14.3.1.2 Convergence of an Infinite Series with Complex Terms 690
• 14.3.1.3 Power Series with Complex Terms 690
3
• 14.3.2 Taylor Series 691
• 14.3.3 Principle of Analytic Continuation 691
• 14.3.4 Laurent Expansion 692
• 14.3.5 Isolated Singular Points and the Residue Theorem 692
4
• 14.3.5.1 Isolated Singular Points 692
• 14.3.5.2 Meromorphic Functions 693
• 14.3.5.3 Elliptic Functions 693
• 14.3.5.4 Residue 693
• 14.3.5.5 Residue Theorem 694
3
2
• 14.4 Evaluation of Real Integrals by Complex Integrals 694
3
• 14.4.1 Application of Cauchy Integral Formulas 694
• 14.4.2 Application of the Residue Theorem 695
• 14.4.3 Application of the Jordan Lemma 695
4
• 14.4.3.1 Jordan Lemma 695
• 14.4.3.2 Examples of the Jordan Lemma 696
3
2
• 14.5 Algebraic and Elementary Transcendental Functions 698
3
• 14.5.1 Algebraic Functions 698
• 14.5.2 Elementary Transcendental Functions 698
• 14.5.3 Description of Curves in Complex Form 701
2
• 14.6 Elliptic Functions 702
3
• 14.6.1 Relation to Elliptic Integrals 702
• 14.6.2 Jacobian Functions 703
• 14.6.3 Theta Function 705
• 14.6.4 Weierstrass Functions 705
2
1
• 15 Integral Transformations 707
2
• 15.1 Notion of Integral Transformation 707
3
• 15.1.1 General Definition of Integral Transformations 707
• 15.1.2 Special IntegralTransformations 707
• 15.1.3 Inverse Transformations 707
• 15.1.4 Linearity of Integral Transformations 707
• 15.1.5 Integral Transformations for Functions of Several Variables 709
• 15.1.6 Applications of Integral Transformations 709
2
• 15.2 Laplace Transformation 710
3
• 15.2.1 Properties of the Laplace Transformation 710
4
• 15.2.1.1 Laplace Transformation, Original and Image Space 710
• 15.2.1.2 Rules for the Evaluation of the Laplace Transformation 711
• 15.2.1.3 Transforms of Special Functions 714
• 15.2.1.4 Dirac 8 Function and Distributions 717
3
• 15.2.2 Inverse Transformation into the Original Space 718
4
• 15.2.2.1 Inverse Transformation with the Help of Tables 718
• 15.2.2.2 Partial Fraction Decomposition 718
• 15.2.2.3 Series Expansion 719
• 15.2.2.4 Inverse Integral 720
3
• 15.2.3 Solution of Differential Equations using Laplace Transformation 721
4
• 15.2.3.1 Ordinary Linear Differential Equations with Constant Coefficients 721
• 15.2.3.2 Ordinary Linear Differential Equations with Coefficients Depending on the Variable 722
• 15.2.3.3 Partial Differential Equations 723
3
2
• 15.3 Fourier Transformation 724
3
• 15.3.1 Properties of the Fourier Transformation 724
4
• 15.3.1.1 Fourier Integral 724
• 15.3.1.2 Fourier Transformation and Inverse Transformation 725
• 15.3.1.3 Rules of Calculation with the Fourier Transformation 727
• 15.3.1.4 Transforms of Special Functions 730
3
• 15.3.2 Solution of Differential Equations using the Fourier Transformation 731
4
• 15.3.2.1 Ordinary Linear Differential Equations 731
• 15.3.2.2 Partial Differential Equations 732
3
2
• 15.4 Z-Transformation 733
3
• 15.4.1 Properties of the Z-Transformation 734
4
• 15.4.1.1 Discrete Functions 734
• 15.4.1.2 Definition of the Z-Transformation 734
• 15.4.1.3 Rules of Calculations 735
• 15.4.1.4 Relation to the Laplace Transformation 736
• 15.4.1.5 Inverse of the Z-Transformation 737
3
• 15.4.2 Applications of the Z-Transformation 738
4
• 15.4.2.1 General Solution of Linear Difference Equations 738
• 15.4.2.2 Second-Order Difference Equations (Initial Value Problem) 739
• 15.4.2.3 Second-Order Difference Equations (Boundary Value Problem) 740
3
2
• 15.5 Wavelet Transformation 740
3
• 15.5.1 Signals 740
• 15.5.2 Wavelets 741
• 15.5.3 Wavelet Transformation 741
• 15.5.4 Discrete Wavelet Transformation 743
4
• 15.5.4.1 Fast Wavelet Transformation 743
• 15.5.4.2 Discrete Haar Wavelet Transformation 743
3
• 15.5.5 Gabor Transformation 743
2
• 15.6 Walsh Functions 744
3
• 15.6.1 Step Functions 744
• 15.6.2 Walsh Systems 744
2
1
• 16 Probability Theory and Mathematical Statistics 745
2
• 16.1 Combinatorics 745
3
• 16.1.1 Permutations 745
• 16.1.2 Combinations 745
• 16.1.3 Arrangements 746
• 16.1.4 Collection of the Formulas of Combinatorics (see Table 16.1) 747
2
• 16.2 Probability Theory 747
3
• 16.2.1 Event, Frequency and Probability 747
4
• 16.2.1.1 Events 747
• 16.2.1.2 Frequencies and Probabilities 748
• 16.2.1.3 Conditional Probability Bayes Theorem 750
3
• 16.2.2 Random Variables. Distribution Functions 751
4
• 16.2.2.1 Random Variable 751
• 16.2.2.2 Distribution Function 751
• 16.2.2.3 Expected Value and Variance, Chebyshev Inequality 753
• 16.2.2.4 Multidimensional Random Variable 754
3
• 16.2.3 Discrete Distributions 754
4
• 16.2.3.1 Binomial Distribution 755
• 16.2.3.2 Hypergeometric Distribution 756
• 16.2.3.3 Poisson Distribution 757
3
• 16.2.4 Continuous Distributions 758
4
• 16.2.4.1 Normal Distribution 758
• 16.2.4.2 Standard Normal Distribution. Gaussian Error Function 759
• 16.2.4.3 Logarithmic Normal Distribution 759
• 16.2.4.4 Exponential Distribution 760
• 16.2.4.5 Weibull Distribution 761
• 16.2.4.6 x2 (Chi-Square) Distribution 762
• 16.2.4.7 Fisher F Distribution 763
• 16.2.4.8 Student t Distribution 763
3
• 16.2.5 Law of Large Numbers. Limit Theorems 764
• 16.2.6 Stochastic Processes and Stochastic Chains 765
4
• 16.2.6.1 Basic Notions. Markov Chains 765
• 16.2.6.2 Poisson Process 768
3
2
• 16.3 Mathematical Statistics 769
3
• 16.3.1 Statistic Function or Sample Function 769
4
• 16.3.1.1 Population. Sample. Random Vector 769
• 16.3.1.2 Statistic Function or Sample Function 770
3
• 16.3.2 Descriptive Statistics 772
4
• 16.3.2.1 Statistical Summarization and Analysis of Given Data 772
• 16.3.2.2 Statistical Parameters 773
3
• 16.3.3 Important Tests 774
4
• 16.3.3.1 Goodness of Fit Test for a Normal Distribution 774
• 16.3.3.2 Distribution of the Sample Mean 776
• 16.3.3.3 Confidence Limits for the Mean 777
• 16.3.3.4 Confidence Interval for the Variance 778
• 16.3.3.5 Structure of Hypothesis Testing 779
3
• 16.3.4 Correlation and Regression 779
4
• 16.3.4.1 Linear Correlation of two Measurable Characters 779
• 16.3.4.2 Linear Regression for two Measurable Characters 780
• 16.3.4.3 Multidimensional Regression 781
3
• 16.3.5 Monte Carlo Methods 783
4
• 16.3.5.1 Simulation 783
• 16.3.5.2 Random Numbers 783
• 16.3.5.3 Example of a Monte Carlo Simulation 784
• 16.3.5.4 Application of the Monte Carlo Method in Numerical Mathematics 785
• 16.3.5.5 Further Applications of the Monte Carlo Method 787
3
2
• 16.4 Calculus of Errors 787
3
• 16.4.1 Measurement Error and its Distribution 788
4
• 16.4.1.1 Qualitative Characterization of Measurement Errors 788
• 16.4.1.2 Density Function of the Measurement Error 788
• 16.4.1.3 Quantitative Characterization of the Measurement Error 790
• 16.4.1.4 Determining the Result of a Measurement with Bounds on the Error 792
• 16.4.1.5 Error Estimation for Direct Measurements with the Same Accuracy 793
• 16.4.1.6 Error Estimation for Direct Measurements with Different Accuracy 793
3
• 16.4.2 Error Propagation and Error Analysis 794
4
• 16.4.2.1 Gauss Error Propagation Law 794
• 16.4.2.2 Error Analysis 796
3
2
1
• 17 Dynamical Systems and Chaos 797
2
• 17.1 Ordinary Differential Equations and Mappings 797
3
• 17.1.1 Dynamical Systems 797
4
• 17.1.1.1 Basic Notions 797
• 17.1.1.2 Invariant Sets 799
3
• 17.1.2 Qualitative Theory of Ordinary Differential Equations 800
4
• 17.1.2.1 Existence of Flows, Phase Space Structure 800
• 17.1.2.2 Linear Differential Equations 801
• 17.1.2.3 Stability Theory 803
• 17.1.2.4 Invariant Manifolds 806
• 17.1.2.5 Poincare Mapping 808
• 17.1.2.6 Topological Equivalence of Differential Equations 810
3
• 17.1.3 Discrete Dynamical Systems 811
4
• 17.1.3.1 Steady States, Periodic Orbits and Limit Sets 811
• 17.1.3.2 Invariant Manifolds 812
• 17.1.3.3 Topological Conjugacy of Discrete Systems 813
3
• 17.1.4 Structural Stability (Robustness) 813
4
• 17.1.4.1 Structurally Stable Differential Equations 813
• 17.1.4.2 Structurally Stable Discrete Systems 814
• 17.1.4.3 Generic Properties 814
3
2
• 17.2 Quantitative Description of Attractors 816
3
• 17.2.1 Probability Measures on Attractors 816
4
• 17.2.1.1 Invariant Measure 816
• 17.2.1.2 Elements of Ergodic Theory 817
3
• 17.2.2 Entropies 819
4
• 17.2.2.1 Topological Entropy 819
• 17.2.2.2 Metric Entropy 819
3
• 17.2.3 Lyapunov Exponents 820
• 17.2.4 Dimensions 822
4
• 17.2.4.1 Metric Dimensions 822
• 17.2.4.2 Dimensions Defined by Invariant Measures 824
• 17.2.4.3 Local Hausdorff Dimension According to Douady and Oesterle 826
• 17.2.4.4 Examples of Attractors 827
3
• 17.2.5 Strange Attractors and Chaos 828
• 17.2.6 Chaos in One-Dimensional Mappings 829
2
• 17.3 Bifurcation Theory and Routes to Chaos 829
3
• 17.3.1 Bifurcations in Morse Smale Systems 829
4
• 17.3.1.1 Local Bifurcations in Neighborhoods of Steady States 830
• 17.3.1.2 Local Bifurcations in a Neighborhood of a Periodic Orbit 835
• 17.3.1.3 Global Bifurcation 838
3
• 17.3.2 Transitions to Chaos 839
4
• 17.3.2.1 Cascade of Period Doublings 839
• 17.3.2.2 Intermittency 839
• 17.3.2.3 Global Homoclinic Bifurcations 840
• 17.3.2.4 Destruction of a Torus 841
3
2
1
• 18 Optimization 846
2
• 18.1 Linear Programming 846
3
• 18.1.1 Formulation of the Problem and Geometrical Representation 846
4
• 18.1.1.1 The Form of a Linear Programming Problem 846
• 18.1.1.2 Examples and Graphical Solutions 847
3
• 18.1.2 Basic Notions of Linear Programming. Normal Form 849
4
• 18.1.2.1 Extreme Points and Basis 849
• 18.1.2.2 Normal Form of the Linear Programming Problem 850
3
• 18.1.3 Simplex Method 851
4
• 18.1.3.1 Simplex Tableau 851
• 18.1.3.2 Transition to the New Simplex Tableau 852
• 18.1.3.3 Determination of an Initial Simplex Tableau 854
• 18.1.3.4 Revised Simplex Method 855
• 18.1.3.5 Duality in Linear Programming 856
3
• 18.1.4 Special Linear Programming Problems 857
4
• 18.1.4.1 Transportation Problem 857
• 18.1.4.2 Assignment Problem 860
• 18.1.4.3 Distribution Problem 860
• 18.1.4.4 Travelling Salesman 861
• 18.1.4.5 Scheduling Problem 861
3
2
• 18.2 Non-linear Optimization 861
3
• 18.2.1 Formulation of the Problem, Theoretical Basis 861
4
• 18.2.1.1 Formulation of the Problem 861
• 18.2.1.2 Optimality Conditions 862
• 18.2.1.3 Duality in Optimization 863
3
• 18.2.2 Special Non-linear Optimization Problems 863
4
• 18.2.2.1 Convex Optimization 863
• 18.2.2.2 Quadratic Optimization 864
3
• 18.2.3 Solution Methods for Quadratic Optimization Problems 865
4
• 18.2.3.1 Wolfe's Method 865
• 18.2.3.2 Hildreth d'Esopo Method 867
3
• 18.2.4 Numerical Search Procedures 867
4
• 18.2.4.1 One-Dimensional Search 867
• 18.2.4.2 Minimum Search in n-Dimensional Euclidean Vector Space 868
3
• 18.2.5 Methods for Unconstrained Problems 868
4
• 18.2.5.1 Method of Steepest Descent (Gradient Method) 869
• 18.2.5.2 Application of the Newton Method 869
• 18.2.5.3 Conjugate Gradient Methods 869
• 18.2.5.4 Method of Davidon, Fletcher and Powell (DFP) 870
3
• 18.2.6 Evolution Strategies 870
4
• 18.2.6.1 Mutation Selection Strategy 871
• 18.2.6.2 Recombination 871
3
• 18.2.7 Gradient Methods for Problems with Inequality Type Constraints) 871
4
• 18.2.7.1 Method of Feasible Directions 872
• 18.2.7.2 Gradient Projection Method 873
3
• 18.2.8 Penalty Function and Barrier Methods 875
4
• 18.2.8.1 Penalty Function Method 875
• 18.2.8.2 Barrier Method 876
3
• 18.2.9 Cutting Plane Methods 877
2
• 18.3 Discrete Dynamic Programming 878
3
• 18.3.1 Discrete Dynamic Decision Models 878
4
• 18.3.1.1 n-Stage Decision Processes 878
• 18.3.1.2 Dynamic Programming Problem 878
3
• 18.3.2 Examples of Discrete Decision Models 879
4
• 18.3.2.1 Purchasing Problem 879
• 18.3.2.2 Knapsack Problem 879
3
• 18.3.3 Bellman Functional Equations 879
4
• 18.3.3.1 Properties of the Cost Function 879
• 18.3.3.2 Formulation of the Functional Equations 880
3
• 18.3.4 Bellman Optimality Principle 880
• 18.3.5 Bellman Functional Equation Method 881
4
• 18.3.5.1 Determination of Minimal Costs 881
• 18.3.5.2 Determination of the Optimal Policy 881
3
• 18.3.6 Examples of Applications of the Functional Equation Method 881
4
• 18.3.6.1 Optimal Purchasing Policy 881
• 18.3.6.2 Knapsack Problem 882
3
2
1
• 19 Numerical Analysis 884
2
• 19.1 Numerical Solution of Non-Linear Equations in a Single Unknown 884
3
• 19.1.1 Iteration Method 884
4
• 19.1.1.1 Ordinary Iteration Method 884
• 19.1.1.2 Newton's Method 885
• 19.1.1.3 Regula Falsi 886
3
• 19.1.2 Solution of Polynomial Equations 887
4
• 19.1.2.1 Horner's Scheme 887
• 19.1.2.2 Positions of the Roots 888
• 19.1.2.3 Numerical Methods 889
3
2
• 19.2 Numerical Solution of Equation Systems 890
3
• 19.2.1 Systems of Linear Equations 890
4
• 19.2.1.1 Triangular Decomposition of a Matrix 890
• 19.2.1.2 Cholesky's Method for a Symmetric Coefficient Matrix 893
• 19.2.1.3 Orthogonalization Method 893
• 19.2.1.4 Iteration Methods 895
3
• 19.2.2 Non-Linear Equation Systems 896
4
• 19.2.2.1 Ordinary Iteration Method 896
• 19.2.2.2 Newton's Method 897
• 19.2.2.3 Derivative-Free Gauss Newton Method 897
3
2
• 19.3 Numerical Integration 898
3
• 19.3.1 General Quadrature Formulas 898
• 19.3.2 Interpolation Quadratures 899
4
• 19.3.2.1 Rectangular Formula 899
• 19.3.2.2 Trapezoidal Formula 899
• 19.3.2.3 Simpson's Formula 900
• 19.3.2.4 Hermite's Trapezoidal Formula 900
3
• 19.3.3 Quadrature Formulas of Gauss 900
4
• 19.3.3.1 Gauss Quadrature Formulas 900
• 19.3.3.2 Lobatto's Quadrature Formulas 901
3
• 19.3.4 Method of Romberg 901
4
• 19.3.4.1 Algorithm of the Romberg Method 901
• 19.3.4.2 Extrapolation Principle 902
3
2
• 19.4 Approximate Integration of Ordinary Differential Equations 904
3
• 19.4.1 Initial Value Problems 904
4
• 19.4.1.1 Euler Polygonal Method 904
• 19.4.1.2 Runge Kutta Methods 904
• 19.4.1.3 Multi-Step Methods 905
• 19.4.1.4 Predictor Corrector Method 906
• 19.4.1.5 Convergence. Consistency. Stability 907
3
• 19.4.2 Boundary Value Problems 908
4
• 19.4.2.1 Difference Method 908
• 19.4.2.2 Approximation by Using Given Functions 909
• 19.4.2.3 Shooting Method 910
3
2
• 19.5 Approximate Integration of Partial Differential Equations 911
3
• 19.5.1 Difference Method 911
• 19.5.2 Approximation by Given Functions 912
• 19.5.3 Finite Element Method (FEM) 913
2
• 19.6 Approximation. Computation of Adjustment, Harmonic Analysis 917
3
• 19.6.1 Polynomial Interpolation 917
4
• 19.6.1.1 Newton's Interpolation Formula 917
• 19.6.1.2 Lagrange's Interpolation Formula 918
• 19.6.1.3 Aitken Neville Interpolation 918
3
• 19.6.2 Approximation in Mean 919
4
• 19.6.2.1 Continuous Problems. Normal Equations 919
• 19.6.2.2 Discrete Problems, Normal Equations, Householder's Method 921
• 19.6.2.3 Multidimensional Problems 922
• 19.6.2.4 Non-Linear Least Squares Problems 922
3
• 19.6.3 Chebyshev Approximation 923
4
• 19.6.3.1 Problem Definition and the Alternating Point Theorem 923
• 19.6.3.2 Properties of the Chebyshev Polynomials 924
• 19.6.3.3 Remes Algorithm 925
• 19.6.3.4 Discrete Chebyshev Approximation and Optimization 926
3
• 19.6.4 Harmonic Analysis 927
4
• 19.6.4.1 Formulas for Trigonometric Interpolation 927
• 19.6.4.2 Fast Fourier Transformation (FFT) 928
3
2
• 19.7 Representation of Curves and Surfaces with Splines 931
3
• 19.7.1 Cubic Splines 931
4
• 19.7.1.1 Interpolation Splines 931
• 19.7.1.2 Smoothing Splines 932
3
• 19.7.2 Bicubic Splines 933
4
• 19.7.2.1 Use of Bicubic Splines 933
• 19.7.2.2 Bicubic Interpolation Splines 933
• 19.7.2.3 Bicubic Smoothing Splines 935
3
• 19.7.3 Bernstein Bezier Representation of Curves and Surfaces 935
4
• 19.7.3.1 Principle of the B B Curve Representation 935
• 19.7.3.2 B B Surface Representation 936
3
2
• 19.8 Using the Computer 936
3
• 19.8.1 Internal Symbol Representation 936
4
• 19.8.1.1 Number Systems 936
• 19.8.1.2 Internal Number Representation 938
3
• 19.8.2 Numerical Problems in Calculations with Computers 939
4
• 19.8.2.1 Introduction, Error Types 939
• 19.8.2.2 Normalized Decimal Numbers and Round-Off 939
• 19.8.2.3 Accuracy in Numerical Calculations 941
3
• 19.8.3 Libraries of Numerical Methods 944
4
• 19.8.3.1 NAG Library 944
• 19.8.3.2 IMSL Library 945
• 19.8.3.3 Aachen Library 946
3
• 19.8.4 Application of Computer Algebra Systems 946
4
• 19.8.4.1 Mathematica 946
• 19.8.4.2 Maple 949
3
2
1
• 20 Computer Algebra Systems 953
2
• 20.1 Introduction 953
3
• 20.1.1 Brief Characterization of Computer Algebra Systems 953
• 20.1.2 Examples of Basic Application Fields 953
4
• 20.1.2.1 Manipulation of Formulas 953
• 20.1.2.2 Numerical Calculations 954
• 20.1.2.3 Graphical Representations 955
• 20.1.2.4 Programming in Computer Algebra Systems 955
3
• 20.1.3 Structure of Computer Algebra Systems 955
4
• 20.1.3.1 Basic Structure Elements 955
3
2
• 20.2 Mathematica 956
3
• 20.2.1 Basic Structure Elements 956
• 20.2.2 Types of Numbers in Mathematica 957
4
• 20.2.2.1 Basic Types of Numbers in Mathematica 957
• 20.2.2.2 Special Numbers 958
• 20.2.2.3 Representation and Conversion of Numbers 958
3
• 20.2.3 Important Operators 959
• 20.2.4 Lists 959
4
• 20.2.4.1 Notions 959
• 20.2.4.2 Nested Lists, Arrays or Tables 960
• 20.2.4.3 Operations with Lists 960
• 20.2.4.4 Special Lists 961
3
• 20.2.5 Vectors and Matrices as Lists 961
4
• 20.2.5.1 Creating Appropriate Lists 961
• 20.2.5.2 Operations with Matrices and Vectors 962
3
• 20.2.6 Functions 963
4
• 20.2.6.1 Standard Functions 963
• 20.2.6.2 Special Functions 963
• 20.2.6.3 Pure Functions 963
3
• 20.2.7 Patterns 964
• 20.2.8 Functional Operations 964
• 20.2.9 Programming 966
• 20.2.10 Supplement about Syntax, Information. Messages 966
4
• 20.2.10.1 Contexts. Attributes 966
• 20.2.10.2 Information 967
• 20.2.10.3 Messages 967
3
2
• 20.3 Maple 968
3
• 20.3.1 Basic Structure Elements 968
4
• 20.3.1.1 Types and Objects 968
• 20.3.1.2 Input and Output 969
3
• 20.3.2 Types of Numbers in Maple 970
4
• 20.3.2.1 Basic Types of Numbers in Maple 970
• 20.3.2.2 Special Numbers 970
• 20.3.2.3 Representation and Conversion of Numbers 970
3
• 20.3.3 Important Operators in Maple 971
• 20.3.4 Algebraic Expressions 971
• 20.3.5 Sequences and Lists 972
• 20.3.6 Tables, Arrays, Vectors and Matrices 973
4
• 20.3.6.1 Tables and Arrays 973
• 20.3.6.2 One-Dimensional Arrays 974
• 20.3.6.3 Two-Dimensional Arrays 974
• 20.3.6.4 Special Commands for Vectors and Matrices 975
3
• 20.3.7 Procedures. Functions and Operators 975
4
• 20.3.7.1 Procedures 975
• 20.3.7.2 Functions 975
• 20.3.7.3 Functional Operators 976
• 20.3.7.4 Differential Operators 977
• 20.3.7.5 The Functional Operator map 977
3
• 20.3.8 Programming in Maple 977
• 20.3.9 Supplement about Syntax, Information and Help 978
4
• 20.3.9.1 Using the Maple Library 978
• 20.3.9.2 Environment Variable 978
• 20.3.9.3 Information and Help 978
3
2
• 20.4 Applications of Computer Algebra Systems 979
3
• 20.4.1 Manipulation of Algebraic Expressions 979
4
• 20.4.1.1 Mathematica 979
• 20.4.1.2 Maple 981
3
• 20.4.2 Solution of Equations and Systems of Equations 984
4
• 20.4.2.1 Mathematica 984
• 20.4.2.2 Maple 986
3
• 20.4.3 Elements of Linear Algebra 988
4
• 20.4.3.1 Mathematica 988
• 20.4.3.2 Maple 989
3
• 20.4.4 Differential and Integral Calculus 992
4
• 20.4.4.1 Mathematica 992
• 20.4.4.2 Maple 995
3
2
• 20.5 Graphics in Computer Algebra Systems 998
3
• 20.5.1 Graphics with Mathematica 998
4
• 20.5.1.1 Basic Elements of Graphics 998
• 20.5.1.2 Graphics Primitives 999
• 20.5.1.3 Syntax of Graphical Representation 999
• 20.5.1.4 Graphical Options 1000
• 20.5.1.5 Two-Dimensional Curves 1002
• 20.5.1.6 Parametric Representation of Curves 1003
• 20.5.1.7 Representation of Surfaces and Space Curves 1003
3
• 20.5.2 Graphics with Maple 1005
4
• 20.5.2.1 Two-Dimensional Graphics 1005
• 20.5.2.2 Three-Dimensional Graphics 1008
3
2
1
• 21 Tables 1010
2
• 21.1 Frequently Used Mathematical Constants 1010
• 21.2 Natural Constants 1010
• 21.3 Metric Prefixes 1012
• 21.4 International System of Physical Units (SI-Units) 1012
• 21.5 Important Series Expansions 1015
• 21.6 Fourier Series 1020
• 21.7 Indefinite Integrals 1023
3
• 21.7.1 Integral Rational Functions 1023
4
• 21.7.1.1 Integrals with X = ax + b 1023
• 21.7.1.2 Integrals with X = ax2 + bx + c 1025
• 21.7.1.3 Integrals with X = a2 � x2 1026
• 21.7.1.4 Integrals with X = a' � x3 1028
• 21.7.1.5 Integrals with X = a4 + x4 1029
• 21.7.1.6 Integrals with X = a4 - x4 1029
• 21.7.1.7 Some Cases of Partial Fraction Decomposition 1029
3
• 21.7.2 Integrals of Irrational Functions 1030
4
• 21.7.2.1 Integrals with pr and a2 � b2x 1030
• 21.7.2.2 Other Integrals with pF 1030
• 21.7.2.3 Integrals with p ax + b 1031
• 21.7.2.4 Integrals with p ax + b and p fx + g 1032
• 21.7.2.5 Integrals with po2 - x2 1033
• 21.7.2.6 Integrals with p x2 + a2 1035
• 21.7.2.7 Integrals with p x2 - a2 1036
• 21.7.2.8 Integrals with ax2 + bx + c 1038
• 21.7.2.9 Integrals with other Irrational Expressions 1040
• 21.7.2.10 Recursion Formulas for an Integral with Binomial Differential 1040
3
• 21.7.3 Integrals of Trigonometric Functions 1041
4
• 21.7.3.1 Integrals with Sine Function 1041
• 21.7.3.2 Integrals with Cosine Function 1043
• 21.7.3.3 Integrals with Sine and Cosine Function 1045
• 21.7.3.4 Integrals with Tangent Function 1049
• 21.7.3.5 Integrals with Cotangent Function 1049
3
• 21.7.4 Integrals of other Transcendental Functions 1050
4
• 21.7.4.1 Integrals with Hyperbolic Functions 1050
• 21.7.4.2 Integrals with Exponential Functions 1051
• 21.7.4.3 Integrals with Logarithmic Functions 1053
• 21.7.4.4 Integrals with Inverse Trigonometric Functions 1054
• 21.7.4.5 Integrals with Inverse Hyperbolic Functions 1055
3
2
• 21.8 Definite Integrals 1056
3
• 21.8.1 Definite Integrals of Trigonometric Functions 1056
• 21.8.2 Definite Integrals of Exponential Functions 1057
• 21.8.3 Definite Integrals of Logarithmic Functions 1058
• 21.8.4 Definite Integrals of Algebraic Functions 1059
2
• 21.9 Elliptic Integrals 1061
3
• 21.9.1 Elliptic Integral of the First Kind F{ip,k), k = sina 1061
• 21.9.2 Elliptic Integral of the Second Kind E (
• 21.9.3 Complete Elliptic Integral, k = sin a 1062
2
• 21.10 Gamma Function 1063
• 21.11 Bessel Functions (Cylindrical Functions) 1064
• 21.12 Legendre Polynomials of the First Kind 1066
• 21.13 Laplace Transformation 1067
• 21.14 Fourier Transformation 1072
3
• 21.14.1 Fourier Cosine Transformation 1072
• 21.14.2 Fourier Sine Transformation 1078
• 21.14.3Fourier Transformation 1083
• 21.14.4Exponential Fourier Transformation 1085
2
• 21.15 Z Transformation 1086
• 21.16 Poisson Distribution 1089
• 21.17 Standard Normal Distribution 1091
• 21.17.1 Standard Normal Distribution for 0.00
• 21.17.2 Standard Normal Distribution for 2.00
• 21.18 x2 Distribution 1093
• 21.19 Fisher F Distribution 1094
• 21.20 Student t Distribution 1096
• 21.21 Random Numbers 1097
1
• 22 Bibliography 1098
2
1
• List of Tables
2
• 1.1 Definition of powers 8
• 1.2 Pascal's triangle 13
• 1.3 Auxiliary values for the solution of equations of degree three 42
• 2.1 Domain and range of trigonometric functions 77
• 2.2 Signs of trigonometric functions 78
• 2.3 Values of trigonometric functions for 0�, 30�, 45�, 60� and 90� 78
• 2.4 Reduction formulas and quadrant relations of trigonometric functions 78
• 2.5 Relations between the trigonometric functions of the same argument in the interval
• 0 < o: < tt/2 80
• 2.6 Domains and ranges of the inverses of trigonometric functions 85
• 2.7 Relations between two hyperbolic functions with the same arguments for x > 0 90
• 2.8 Domains and ranges of the area functions 92
• 2.9 For the approximate determination of an empirically given function relation 112
• 3.1 Names of angles in degree and radian measure 129
• 3.2 Properties of some regular polygons 139
• 3.3 Defining quantities of a right angled-triangle in the plane 141
• 3.4 Defining quantities of a general triangle, basic problems 144
• 3.5 Conversion between Degrees and Gons 145
• 3.6 Directional angle in a segment with correct sign for arctan 145
• 3.7 Regular polyeders with edge length a 154
• 3.8 Defining quantities of a spherical right-angled triangle 168
• 3.9 First and second basic problems for spherical oblique triangles 170
• 3.10 Third basic problem for spherical oblique triangles 171
• 3.11 Fourth basic problem for spherical oblique triangles 172
• 3.12 Fifth and sixth basic problemes for a spherical oblique triangle 173
• 3.13 Scalar product of basis vectors 186
• 3.14 Vector product of basis vectors 186
• 3.15 Scalar product of reciprocal basis vectors 186
• 3.16 Vector product of reciprocal basis vectors 186
• 3.17 Vector equations 188
• 3.18 Geometric application of vector algebra 189
• 3.19 Equation of curves of second order. Central curves ((5 ^ 0) 205
• 3.20 Equations of curves of second order. Parabolic curves (5 = 0) 206
• 3.21 Coordinate signs in the octants 208
• 3.22 Connections between Cartesian, cylindrical, and spherical polar coordinates 210
• 3.23 Notation for the direction cosines under coordinate transformation 211
• 3.24 Type of surfaces of second order with 5 ^ 0 (central surfaces) 224
• 3.25 Type of surfaces of second order with 5 = 0 (paraboloid, cylinder and two planes) 224
• 3.26 Tangent and normal equations 226
• 3.27 Vector and coordinate equations of accompanying configurations of a space curve 241
• 3.28 Vector and coordinate equations of accompanying configurations as functions of the arclength 241
• 3.29 Equations of the tangent plane and the surface normal 246
• 5.1 Truth table of propositional calculus 286
• 5.2 NAND function 288
• 5.3 NOR function 288
• 5.4 Primitive Bravais lattice 310
• 5.5 Bravais lattice, crystal systems, and crystallographic classes 311
• 5.6 Some Boolean functions with two variables 344
• 5.7 Tabular representation of a fuzzy set 360
• 5.8 t- and s-norms, p G IR 367
• 5.9 Comparison of operations in Boolean logic and in fuzzy logic 369
• 6.1 Derivatives of elementary functions 381
• 6.2 Differentiation rules 386
• 6.3 Derivatives of higher order of some elementary functions 387
• 7.1 The first Bernoulli numbers 412
• 7.2 First Euler numbers 413
• 7.3 Approximation formulas for some frequently used functions 419
• 8.1 Basic integrals 428
• 8.2 Important rules of calculation of indefinite integrals 430
• 8.3 Substitutions for integration of irrational functions I 435
• 8.4 Substitutions for integration of irrational functions II 436
• 8.5 Important properties of definite integrals 443
• 8.6 Line integrals of the first type 465
• 8.7 Curve elements 465
• 8.8 Plane elements of area 474
• 8.9 Applications of the double integral 475
• 8.10 Elementary volumes 479
• 8.11 Applications of the triple integral 480
• 8.12 Elementary regions of curved surfaces 482
• 11.1 Roots of the Legendre polynomial of the first kind 574
• 13.1 Relations between the components of a vector in Cartesian, cylindrical, and spherical coordinates 648
• 13.2 Fundamental relations for spatial differential operators 658
• 13.3 Expressions of vector analysis in Cartesian, cylindrical, and spherical coordinates 659
• 13.4 Line, surface, and volume elements in Cartesian, cylindrical, and spherical coordinates 660
• 14.1 Real and imaginary parts of the trigonometric and hyperbolic functions 700
• 14.2 Absolute values and arguments of the trigonometric and hyperbolic functions 700
• 14.3 Periods, roots and poles of Jacobian functions 704
• 15.1 Overview of integral transformations of functions of one variable 708
• 15.2 Comparison of the properties of the Fourier and the Laplace transformation 730
• 16.1 Collection of the formulas of combinatorics 747
• 16.2 Relations between events 748
• 16.3 Frequency table 773
• 16.4 x2 test 776
• 16.5 Confidence level for the sample mean 777
• 16.6 Error description of a measurement sequence 794
• 17.1 Steady state types in three-dimensional phase space 809
• 19.1 Helping table for FEM 916
• 19.2 Orthogonal polynomials 920
1
• List of Tables
2
• 19.3 Number systems 937
• 19.4 Parameters for the basic forms 939
• 19.5 Mathematica, numerical operations 946
• 19.6 Mathematica, commands for interpolation 947
• 19.7 Mathematica, numerical solution of differential equations 948
• 19.8 Maple, options for the command fsolve 950
• 20.1 Mathematica, Types of numbers 957
• 20.2 Mathematica, Important operators 959
• 20.3 Mathematica, Commands for the choice of list elements 960
• 20.4 Mathematica. Operations with lists 960
• 20.5 Mathematica, Operation Table 961
• 20.6 Mathematica, Operations with matrices 962
• 20.7 Mathematica, Standard functions 963
• 20.8 Mathematica, Special functions 963
• 20.9 Maple, Basic types 968
• 20.10 Maple, Types 968
• 20.11 Maple, Types of numbers 970
• 20.12 Maple, Arguments of function convert 971
• 20.13 Maple, Standard functions 975
• 20.14 Maple, Special functions 976
• 20.15 Mathematica, Commands for manipulation of algebraic expressions 979
• 20.16 Mathematica, Algebraic polynomial operations 980
• 20.17 Maple, Operations to manipulate algebraic expressions 981
• 20.18 Mathematica, Operations to solve systems of equations 986
• 20.19 Maple, Matrix operations 990
• 20.20 Maple, Operations of the Gaussian algorithm 991
• 20.21 Mathematica, Operations of differentiation 992
• 20.22 Mathematica, Commands to solve differential equations 994
• 20.23 Maple, Options of operation dsolve 997
• 20.24 Mathematica, Two-dimensional graphic objects 999
• 20.25 Mathematica, Graphics commands 999
• 20.26 Mathematica, Some graphical options 1000
• 20.27 Mathematica, Options for 3D graphics 1005
• 20.28 Maple, Options for Plot command 1006
• 20.29 Maple, Options of command plot3d 1008
• 21.1 Frequently Used Constants 1010
• 21.2 Natural Constants 1010
• 21.3 Metric Prefixes 1012
• 21.4 International System of Physical Units (SI-Units) 1012
• 21.5 Important Series Expansions 1015
• 21.6 Fourier Series 1020
• 21.7 Indefinite Integrals 1023
• 21.8 Definite Integrals 1056
• 21.9 Elliptic Integrals 1061
• 21.10 Gamma Function 1063
• 21.11 Bessel Functions (Cylindrical Functions) 1064
• 21.12 Legendre Polynomials of the First Kind 1066
• 21.13 Laplace Transformation 1067
• 21.14 Fourier Transformation 1072
• 21.15 Z- Transformation 1086
• 21.16 Poisson Distribution 1089
1
• List of Tables XLIII
2
• 21.17 Standard Normal Distribution 1091
• 21.18 x2 Distribution 1093
• 21.19 Fisher F Distribution 1094
• 21.20 Student t Distribution 1096
• 21.21 Random Numbers 1097
1
• Index 1109
• Mathematic Symbols 1160

0

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1
• A
2
• A
• Abel --- rectangle, square,
3
• Abel
• --- group, 299
• --- integral equation, 590
• --- theorem, 416
• abscissa
• --- plane coordinates, 189
• --- space coordinates, 208
• absolute term, 272
• absolutely
• --- continuous, 638
• --- convergent, 455
• --- integrable, 455
• absorbing set, 799
• absorption law
• --- Boolean algebra, 342
• --- propositional logic, 287
• --- sets, 293
• account number system, 333
• --- uniform, 333
• accumulation factor, 22
• accumulation point. 606
• accuracy, measure of, 789
• addition
• --- complex numbers, 36
• --- computer calculation, 940
• --- polynomials, 11
• --- rational numbers, 1
• addition theorem
• area functions, 93
• --- hyperbolic functions, 89
• --- inverse trigonometric functions, 86
• --- trigonometric functions, 79
4
• 81
3
• additivity, (7-additivity, 635
• adjacency, 348
• --- matrix, 350
• adjacent side, 130
• adjoint, 259
• admittance matrix, 355
• a.e. (almost everywhere), 636
• AES algorithm (Advanced Encryption Standard), 339
• aggregation operator, 370
• algebra, 251
4
• 286
3
• --- Boolean, 342
4
• 747
3
• --- finite, 343
• --- classical structures, 298
• --- commutative, 614
• --- factor algebra. 340
• --- free, 341
• --- linear, 251
• --- normed, 614
• --- ft-algebra, 340
• --- ft-subalgebra, 340
• --- (T-algebra, 635
• --- Borelian, 636
• --- switch algebra, 342
4
• 346
3
• --- term algebra, 341
• --- universal algebra, 340
• algorithm
• AES (Advanced Encryption Standard), 339
• Aitken-Neville, 919
• --- Dantzig, 357
• --- Euclidean, 3
4
• 14
• 323
3
• --- polynomial rings, 315
• --- theorem, 324
• --- Ford-Fulkerson, 359
• --- Gauss, 276
4
• 891
3
• --- graph theory, 348
• --- Kruskal, 355
• --- maximum flow, 359
• --- QR, 283
• --- Rayleigh-Ritz, 283
• --- Remes, 925
• --- Romberg method, 901
• alignment
• --- chart, 123
4
• 124
3
• --- nomogram, 125
• --- Q-cut, 364
• --- strong, 364
• --- Q-level set, 364
• --- strong, 364
• --- Q-limit set, 799
4
• 805
• 812
3
• alternating point, 923
• --- theorem, 923
• alternating tensor, 265
• altitude
• --- cone, 156
• --- cylinder, 154
• --- polyhedron, 151
• --- triangle, 132
• amortization calculus, 23
• amplitude
• --- function, 703
• --- sine function, 75
• --- spectrum, 727
• analysis
• --- functional, 596
• --- harmonic, 917
4
• 927
3
• --- multi-scale analysis, 743
• --- numerical, 884
• --- vector, 642
• analytical geometry
• --- plane, 189
• --- space, 207
• angle, 128
• acute, 129
• --- between vectors, 189
• --- central, 130
4
• 140
3
• --- chord and tangent, 139
• --- circumference, 139
• --- convex, 129
• --- directional, 145
• --- escribed circle, 140
• --- Eulerian, 211
• --- exterior, 139
• --- full, 129
• --- geodesy, 145
• --- inclination, 379
• --- interior, 139
• --- line and plane, 219
• --- lines, plane, 128
• angle (continued)
• --- lines, space, 219
• --- measure in degrees, 130
• --- names, 128
• --- notion, 128
• --- nutation, 211
• --- obtuse, 129
• --- perigon, 130
• --- plane, 151
• --- notion, 128
• --- plane curves, 228
• --- planes, space, 216
• --- precession, 211
• --- radian measure, 130
• --- reduction, 174
• --- right, 129
• --- round, 129
• --- secant and tangent, 140
• --- slope, 379
• --- tangent, 227
• --- solid, 151
• --- space curves, 246
• --- straight, 129
• --- tilt, 143
• angles
• adjacent, 129
• alternate, 129
• at parallels, 129
• --- complementary, 129
• --- corresponding, 129
• --- exterior-interior, 129
• --- opposite, 129
• --- sum
• --- plane triangle, 132
• --- spherical triangle, 163
• --- supplementary, 129
• --- vertex, 129
• angular
• --- coefficient, curve third degree, 66
• --- coefficient, plane, 194
• --- frequency, 82
• annuity, 23
4
• 24
3
• --- calculation, 25
• --- payment, 25
• --- perpetual, 25
• annulator, 624
• annulus, 140
• area, 140
• Anosov diffeomorphism, 829
• anticommutativity, vector product, 183
• antiderivative, 427
• antikink soliton, 550
• antilog, 10
• antisoliton, 549
• Apollonius
• --- circle, 683
• --- theorem, 199
• apothem, 132
• applicate, 208
• approximate equation, 496
• approximate formula, empirical curve, 106
• approximation, 917
• asymptotic, polynomial part, 15
• --- best, Hilbert space, 617
• --- bicubic, 935
• --- Chebyshev, 923
• --- i5 function, 717
• --- formulas
• --- series expansion, 418
• --- in mean, 403
4
• 919
3
• --- Liouville theorem, 4
• --- numbers, 4
• --- partial differential equations, 912
• --- problem, 616
• --- solution by extreme value, 403
• --- successive
• --- Banach space, 621
• --- differential equations, ordinary, 496
• --- integral equation, 567
• --- uniform, 923
• --- using given functions, 909
• --- Weierstrass theorem, 607
• arc
• --- ellipse, 200
• --- graph,348
• --- chain, 357
• --- hyperbola, 203
• --- intersection, 149
• --- length
• --- circular segment, 140
• --- line integral, first type, 464
• --- parabola, 204
• --- plane curve, 140
4
• 449
3
• --- space curve, 246
4
• 464
3
• --- sequences, 357
• Archimedean spiral, 103
• area
• annulus, 140
• --- sector, 140
• --- circle, 139
• --- circular sector, 140
• --- circular segment, 140
• --- cosine, 91
• --- cotangent, 92
• --- curved surface, 482
• --- curvilinear bounded, 448
• --- curvilinear sector, 448
• --- double integral, 474
• --- ellipse, 199
• --- formula, Heron's, 143
• --- function, 91
• --- hyperbola, 202
• --- parabola, 204
• --- parallelogram, 134
• --- parallelogram with vectors, 189
• --- planar figures, 448
• --- polyeder with vectors, 189
• --- polygon, 193
• --- rectangle, square, 135
2
• rhombus, --- azimuthal equation,
3
• --- rhombus, 135
• --- similar plane figures, 133
• --- sine, 91
• --- subset, 635
• --- surface patch, 247
• --- tangent, 92
• --- triangle, 193
• area (continued)
• --- plane, 141
4
• 143
3
• --- spherical, 163
4
• 167
3
• argument
• --- function of one variable, 47
• --- function of several variables, 117
• arithmetic, 1
• --- sequence, 18
• Arnold tongue, 844
• arrangement, 745
4
• 746
3
• --- with repetition, 746
• --- without repetition, 746
• arrow
• --- diagram, 294
• --- function, 294
• article number, European, 333
• ASCII (American Standard Code for Information Inte
• --- change), 936
• assignment problem, 860
• associative law
• --- Boolean algebra, 342
• --- matrices, 254
• --- propositional logic, 287
• --- sets, 293
• --- tensors, 264
• associativity, product of vectors, 184
• astroid, 102
4
• 474
3
• asymptote
• --- curve, 231
4
• 234
3
• --- definition, 234
• attractor, 800
• --- chaotic, 828
• --- fractal, 828
• --- Henon, 826
4
• 828
3
• --- hyperbolic, 828
• --- Lorenz, 827
• --- strange, 828
• autocorrelation function, 818
• axial field, 644
• axioms
• algebra, 614
• --- closed set, 606
• --- metric spaces, 604
• --- normed space, 611
• --- open set, 606
• --- ordered vector space, 601
• --- pseudonorm, 624
• --- scalar product, 615
• --- vector space, 596
• axis
• abscissae
• --- plane coordinates, 189
• --- space coordinates, 208
• --- ordinates
• --- plane coordinates, 189
• --- space coordinates, 208
• --- parabola, 203
• azimuth, 159
• azimuthal equation, 543
2
1
• B
2
• B
• backward substitution, 891
• Baire, second Baire category, 814
• Bairstow method, 889
• ball, metric space, 605
• Banach
• --- fixed-point theorem, 631
• --- space, 612
• --- example, 612
• --- series, 612
• --- theorem, continuity, inverse operator, 621
• band structure, coefficient matrix, 895
• barrel, 157
• --- circular, 157
• --- parabolic, 157
• base, 7
• --- power, 7
• --- vector, 186
• --- reciprocal, 185
3
• 186
2
• --- vector space, 317
• basic
• --- formulas
• --- plane trigonometry, 141
• --- problems
• --- plane trigonometry, 143
• --- spherical trigonometry, 167
• --- theorems
• --- propositional logic, 287
• basis, 599
• --- contravariant, 266
• --- covariant, 266
• --- inverse, 851
• --- vector
• --- contravariant, 266
• --- covariant, 266
• --- vector space, 599
• Bayes theorem, 750
• B-B representation
• --- curve, 935
• --- surface, 935
• Berge theorem, 356
• Bernoulli
• --- inequality, 30
• --- numbers, 412
• --- shift, 829
• Bernoulli-lHospital rule, 54
• Bernstein polynomial, 935
• Bessel
• --- differential equation, 509
• --- differential equation, linear, zero order, 722
• --- function, 509
• --- imaginary variables, 509
• --- modified, 509
• --- inequality, 618
• beta function, 1056
• biangle, spherical, 161
• bifurcation
• Bogdanov-Takens, 834
• --- codimension, 829
• --- cusp, 833
• --- flip bifurcation, 836
• --- global, 829
3
• 838
2
• --- homoclinic, 840
• --- Hopf bifurcation, 831
• --- generalized, 834
• --- local, 829
• --- mappings, subcritical saddle node, 836
• bifurcation (continued)
• --- pitchfork, 836
• --- supercritical, 834
• --- saddle node, 831
• --- transcritical, 831
• binary number, 816
3
• 937
2
• binary system, 937
• binomial, 69
• --- coefficient, 13
• --- distribution, 755
• --- formula, 12
• --- linear, 69
• --- quadratic, 69
• --- theorem, 12
• binormal, space curve, 239
3
• 241
2
• biquadratic equation, 42
• birth process, 769
• bisection method, 283
• bisector, 142
• --- triangle, 132
• bit, 936
• --- reversing the order, 930
• block, 152
• body of revolution, lateral surface, 449
• Bolzano theorem
• --- one variable, 59
• --- several variables, 123
• Bolzano-Weierstrass property, 628
• Boolean
• --- algebra. 342
3
• 747
2
• --- finite, 343
• --- expression, 344
• --- function, 287
3
• 344
2
• --- n-ary, 344
• --- variable, 344
• Borel
• --- set, 636
• --- a-algebra, 636
• bound
• --- function, 50
• --- sequence, 404
• boundary
• --- collocation, 913
• --- condition, 552
• --- uncertain for some variables (fuzzy), 369
• boundary value
• --- conditions, 487
3
• 514
2
• --- problem, 487
3
• 514
• 908
2
• --- Hilbert, 593
• --- Hilbert, homogeneous, 593
• --- homogeneous, 514
• --- inhomogeneous, 514
• --- linear, 514
• bounded set (order-bounded), 602
• boundedness of a function
• --- one variable, 60
• --- several variables, 123
• Bravais lattice, 310
• break of symmetry, bifurcation, 834
• Breit-Wigner curve, 731
• Brouwer fixed-point theorem, 633
• business mathematics, 21
• byte, 936
1
• C
2
• C
• calculation --- max-average,
3
• calculation
• complex numbers, 36
• --- determinants, 260
• --- numerical
• --- accuracy, 941
• --- basic operations, 940
• --- tensors, 263
• calculus
• --- differentiation, 379
• --- errors, 787
• --- integral, 427
• --- observations, measurement error, 787
• --- propositional, 286
• --- variations, 552
• canonical form, circle-mapping, 843
4
• 844
3
• canonical system, 519
• Cantor function, 844
• Cantor set, 822
4
• 823
• 827
3
• cap, spherical, 157
• capacity, edge, 358
• Caratheodory condition, 631
• Cardano formula, 41
• cardinal number, 291
4
• 298
3
• cardinality, set, 298
• cardioid, 98
• carrier function, 575
• Carson transformation, 707
• Cartesian
• coordinates
• --- plane, 189
• --- space, 208
• --- folium, 94
• cartography scale, 144
• cascade, period doubling, 839
4
• 842
3
• Cassinian curve, 98
• category, second Baire category, 814
• catenary curve, 88
4
• 105
• 555
3
• catenoid, 88
• Cauchy
• --- integral, 592
• --- integral formula, 689
• --- integral formulas, application, 694
• --- method, differential equations of higher order, 502
• --- principal value
• --- improper integral, 457
• --- principal value, improper integral, 454
• --- principle, 608
• --- problem, 518
• --- sequence, 607
• --- theorem, 390
• Cauchy-Riemann differential equations, partial. 672
• Cayley
• --- table, 300
• --- theorem, 302
4
• 355
3
• center
• circle, 197
• curvature, 230
• --- spherical, 175
• center manifold theorem
• --- differential equations, 830
• --- mappings, 835
• center of area method, 374
• center of gravity, 213
• --- arbitrary planar figure. 453
• --- arc segment, 452
• closed curve, 452
• --- double integral, 474
• --- line integral, first type, 464
• --- method, 373
• --- generalized, 374
• --- parametrized. 374
• --- trapezoid, 453
• --- triangle, 132
• --- triple integral, 480
• center of mass, 191
4
• 213
3
• central
• --- angle, 130
• curves, 205
• --- field, 643
• --- limit theorem, Lindeberg�Levy, 765
• --- surface, 220
• centroid, points having masses, 191
• chain, 297
4
• 351
3
• --- directed, elementary, 357
• --- graph,357
• --- elementary, 357
• --- Markov, 766
• --- stationary, 766
• --- time-homogeneous, 766
• --- rule, 643
• composite function, 382
• --- stochastic, 765
4
• 766
3
• chaos, 797
4
• 841
3
• --- attractor, strange, 828
• --- from torus to chaos, 841
• --- one-dimensional mapping, 829
• --- routes to chaos, 829
• --- through intermittence, 843
• --- transition to chaos, 839
• character
• --- group element, 304
• --- representation of groups, 305
• characteristic, 10
• characteristic equation, 494
4
• 503
• 505
3
• characteristic strip, 519
• Chebyshev
• --- approximation, 923
• continuous, 923
• --- discrete, 926
• --- formula, 87
• --- inequality, 31
• --- polynomials, 924
• --- theorem, 436
• Chinese remainder theorem, 328
• --- X2 distribution, 762
• --- X2 test, 775
4
• 776
3
• Cholesky
• --- decomposition, 893
• --- method, 278
4
• 893
3
• chord, 140
• --- theorem, 138
• circle
• --- Apollonius, 682
• --- area, 138
• center, 197
• chord, 138
• circumference, 138
• convergence, 690
• curvature, 230
• --- plane curve, 228
• --- dangerous, 149
• --- definition, 138
4
• 197
3
• --- equation
• Cartesian coordinates, 197
• --- polar coordinates, 197
• --- great circle, 158
4
• 173
3
• --- intersection, 158
• --- mapping, 843
4
• 844
3
• --- parametric representation, 197
• --- periphery, 138
• --- plane, 138
4
• 197
3
• --- radius, 138
4
• 197
3
• --- small circle, 158
4
• 175
3
• --- tangent, 138
4
• 198
3
• circuit
• --- directed, graph, 357
• --- Euler, 352
• --- Hamilton, 353
• circuit integral, 468
• --- being zero, 470
• circular
• --- field, 646
• --- point, 249
• --- sector, 140
• --- segment, 140
• circumcircle
• --- quadrangle, 136
• --- triangle, 132
4
• 142
3
• circumscribing
• --- quadrangle, 136
• --- triangle, 132
• cissoid, 95
• Clairaut
• --- differential equation, ordinary, 493
• --- differential equation, partial, 520
• class
• --- defined by identities, 341
• --- equivalence class, 297
• --- midpoint, 773
• --- statistics, 772
• Clebsch-Gordan
• coefficient, 306
• --- series, 306
• --- theorem, 306
• closure
• closed linear, 616
• --- linear, 597
• --- set, 607
• --- transitive, 295
• clothoid, 105
• code, 331
• --- ASCII (American Standard Code for Information
• --- Interchange), 936
• --- public key, 331
• --- RSA (Rivest, Shamir, Adleman), 331
• codimension, 829
• coding, 331
4
• 335
3
• coefficient, 11
• Clebsch-Gordan, 306
• coefficient (continued)
• comparison. 16
• --- Fourier, 420
• --- leading, 38
• --- matrix, extended, 892
• --- metric, 185
4
• 186
3
• --- vector decomposition, 182
• collinearity, vectors, 184
• collocation
• --- boundary, 913
• --- domain. 913
• --- method, 576
4
• 909
• 913
3
• --- points, 909
4
• 913
3
• column pivoting, 892
• column sum criterion, 895
• combination, 745
• --- with repetition, 745
• --- without repetition, 745
• combinatorics, 745
• commensurability, 4
• commutative law
• --- Boolean algebra, 342
• --- matrices, 254
• --- propositional logic, 287
• --- sets, 293
• --- vectors, 255
• commutativity, scalar product, 183
• commutator, 319
• comparable function, 558
• complement, 292
• --- algebraic, 259
• --- fuzzy, 368
• --- fuzzy set, 365
• --- orthogonal, 616
4
• 624
3
• --- sets, 292
• --- Sugeno complement, 368
• --- Yager complement, 368
• complementary angles formulas, 78
• completeness relation, 618
• completion, metric space, 610
• complex analysis, 671
• complex function, 47
• --- pole, 673
• complex number, 34
• --- plane, 34
• complex mapping, 684
• complex-valued function, 47
• complexification, 601
• composition, 371
• --- max-average, 371
2
• max-(/-norm), --- law,
3
• --- max-(/-norm), 371
• compound interest, 22
• calculation, 22
• computation of adjustment, 917
• computer
• --- basic operations, 940
• --- error of the method, 942
• --- error types, 939
4
• 941
3
• --- internal number representation, 938
• --- internal symbol representation, 936
• --- numerical accuracy, 941
• --- numerical problems in calculation, 939
• --- use of computers, 936
• computer algebra systems
• --- applications, 979
• --- basic structure elements, 955
• --- differential and integral calculus, 992
• --- elements of linear algebra, 988
• --- equations and systems of equations, 984
• --- functions, 956
• --- graphics, 998
• --- infix form, 956
• --- list, 956
• --- manipulation of algebraic expressions, 979
• --- numbers, 955
• --- object, 955
• --- operator, 956
• --- prefix form, 956
• --- programming, 955
• --- purpose, 953
• --- set, 956
• --- suffix form, 956
• --- terms, 956
• --- type, 955
• --- variable, 956
• concave, curve, 228
• conchoid
• --- Nicomedes, 96
• --- general, 96
• --- of the circle, 96
• --- of the line, 96
• conclusion, 372
• concurrent expressions, 345
• condition
• --- boundary value, ordinary differential equation, 487
• Caratheodory, 631
• Cauchy (convergence), 122
• --- Dirichlet (convergence), 422
• --- initial value, ordinary differential equation, 487
• --- Kuhn-Tucker, 862
• --- Lipschitz, higher-order differential equation, 498
• --- Lipschitz, ordinary differential equation, 488
• --- number, 894
• --- regularity condition, 876
• --- Slater, 863
• cone, 156
4
• 221
• 613
3
• central surface, 224
• circular, 156
• --- generating, 602
• --- imaginary, 224
• --- normal, 613
• --- regular, 613
• --- solid, 613
• --- truncated, 156
• --- vector space, 599
• confidence
• --- interval, 781
• --- regression coefficient, 781
• --- variance, 778
• --- limit
• --- for the mean, 777
• --- prescription, 792
• --- probability, 776
• --- region, 781
• congruence
• --- algebraic, 327
• corners, 151
• --- linear, 328
• --- method, 783
• --- plane figures, 133
• --- polynomial congruence. 330
• --- quadratic, 329
• --- relation, 340
• --- kernel, 341
• --- simultaneous linear, 328
• --- system simultaneous linear, 328
• --- theorems, 133
• congruent
• --- directly, 132
• --- indirectly, 133
• --- mapping, 132
4
• 133
3
• conic section, 156
4
• 205
• 207
3
• --- singular, 207
• conjugate complex number, 36
• conjunction, 286
• --- elementary, 345
• consistency
• --- integration of differential equation, 907
• --- order p, 907
• constant
• --- Euler, 460
• --- in polynomial, 60
• --- propositional calculus, 286
• --- term, 272
• constants
• --- mathematical, frequently used, 1010
• --- natural, 1010
• continuation, analytic, 691
• continued fractions, 3
• continuity
• composite functions, 59
• --- elementary function, 58
• --- from below, 635
• --- function
• --- one variable, 57
• --- several variables, 122
• --- Holder, 591
• continuous, absolutely, 638
• continuum, 298
• contour integral
• --- vector field, 662
• contracting principle, 608
• contraction, 264
• --- tensor, 268
• contradiction, Boolean function, 344
• control digit, 332
• convergence
• --- absolute, 409
4
• 416
3
• complex terms, 690
• --- alternating series test of Leibniz, 410
• --- Banach space, 612
• circle of convergence, 690
• condition of Cauchy, 52
• conditional, 409
• complex terms, 690
• --- in mean, 422
• --- infinite series, complex terms, 690
• --- integration of differential equation, 907
• --- non-uniform, 414
• --- order p, 907
• --- sequence of numbers, 405
• complex terms, 689
• --- series, 406
4
• 408
3
• complex terms, 689
• --- uniform, 414-416
• --- uniformly, function sequences, 606
• --- weak,629
• --- Weierstrass criterion, 415
• convergence criterion
• comparison criterion, 407
• --- D'Alembert's ratio test, 407
• --- integral test of Cauchy, 408
• --- root test of Cauchy, 408
• convergence theorem, 406
• --- measurable function, 638
• conversion, number systems, 937
• convex, curve, 228
• convolution
• --- Fourier transformation, 729
• --- Laplace transformation, 713
• --- one-sided, 729
• --- two-sided, 729
• --- z-transformation, 736
• coordinate inversion, 269
• coordinate line, 209
4
• 266
3
• coordinate surface, 209
4
• 266
3
• coordinate system
• --- double logarithmic, 115
• --- Gauss-Krueger, 143
• --- left-hand, 207
• --- orientation, 207
• --- orthogonal, 180
• --- orthonormal, 180
• --- plane, 189
• --- right-hand, 207
• --- semilogarithmic. 115
• --- Soldner, 143
• --- spatial, 207
• --- transformation, 262
• coordinate transformation, 210
4
• 267
• 647
3
• --- equation
• central curves second order, 205
• --- quadratic curve (parabolic), 206
• coordinates
• --- affine, 185
• --- axis, 189
• --- barycentric, 917
• Cartesian, 186
• --- plane, 189
• --- space, 208
• Cartesian, transformation, 191
• contravariant, 187
4
• 268
3
• covariant, 187
4
• 268
3
• curvilinear, 190
4
• 244
• 266
3
• --- three dimensional, 209
• cylindrical, 209
• --- vector field, 647
• --- Descartes, 189
• --- equation, space curve, 241
• --- Gauss, 244
• --- Gauss-Kruger, 160
• --- geodetic, 143
• ccordinates (continued)
• --- geographical, 160
• --- mixed, 267
• --- point, 189
• --- polar, plane, 190
• --- polar, spherical, 209
• --- representation with scalar product, 187
• --- Soldner, 160
• --- spherical, 209
• --- vector field, 647
• --- transformation, 190
• --- triangle coordinates, 917
• --- vector, 182
• coprime, 5
• corner
• convex, 151
• --- figure, 150
• --- northwest corner rule, 858
• --- symmetric, 151
• --- trihedral, 151
• correction form, 896
• corrector, 906
• correlation, 779
• --- analysis, 779
• coefficient, 780
• --- empirical, 780
• cosecant
• --- geometric definition, 130
• --- hyperbolic, 87
• --- trigonometric, 76
• coset
• --- left, 301
• --- right, 301
• cosine
• --- geometric definition, 130
• --- hyperbolic, 87
• --- geometric definition. 131
• --- law, 142
2
• law for sides, --- cylindrical surface,
3
• --- law for sides, 164
• --- rule, spherical triangle, 164
• --- trigonometric, 75
• cotangent
• --- geometric definition, 130
• --- hyperbolic, 87
• --- trigonometric, 76
• Coulomb field, point-like charge, 646
4
• 668
3
• counterpoint, 158
• course angle, 159
• covariance, two-dimensional distribution, 780
• covering transformation, 299
• covering, open, 800
• Cramer rule, 275
• credit, 22
• criterion
• convergence
• --- sequence of numbers, 405
• --- series, 406
• --- series with positive terms, 407
• --- uniform, Weierstrass, 415
• --- divisibility, 323
• --- subspace, 317
• cross product, 183
• cryptanalysis, classical, methods, 337
• --- Kasiski-Friedman test, 337
• --- statistical analysis, 337
• cryptography, conventional
• --- methods, 335
• --- linear substitution ciphers, 336
• --- substitution, 335
• --- monoalphabetic, 335
• --- monographic, 335
• --- polyalphabetic, 335
• --- polygraphic 335
• --- transposition, 335
• cryptology, 334
• --- AES algorithm, 339
• classical
• --- Hill cypher method, 336
• --- matrix substitution method, 336
• --- Vigenere cypher method, 336
• cryptosystem, 334
• --- Diffie-Hellman key exchange, 338
• --- encryption
• context free, 334
• context sensitive, 334
• --- IDEA algorithm, 340
• --- mathematical foundation, 334
• --- one-time pad, 338
• --- one-way function, 339
• --- public key methods, 338
• --- RSA method, 339
• --- security of cryptosystems, 335
• --- subject, 334
• crystal class, 311
• crystal system, 311
• crystallography
• --- lattice, 310
• --- symmetry group, 310
• cube, 152
• curl, 668
• --- density. 669
• --- field
• --- pure, 668
• --- zero-divergence field, 668
• --- line, 655
• curvature
• center, 230
• circle, 230
• curves on a surface, 247
• --- Gauss surface, 249
• --- mean, surface, 249
• --- minimal total curvature, 932
• --- plane curve, 228
• --- radius, 230
• curve on a surface, 247
• --- principal, 247
• --- space curve, 240
• --- surface, 247
4
• 249
3
• constant curvature, 249
• --- total, 243
• curve
• --- algebraic, 93
4
• 194
3
• --- n-th order, 234
• --- arc cosine, 84
• --- arc cotangent, 84
• --- arc sine, 84
• curve (continued I)
• --- arc tangent, 84
• --- Archimedean spiral, 103
• --- area cosine, 91
• --- area cotangent, 92
• --- area sine, 91
• --- area tangent, 92
• --- astroid, 102
• --- asymptote, 231
4
• 234
3
• --- asymptotic point. 232
• --- B-B representation, 935
• cardioid, 98
• Cartesian folium, 94
• Cassinian, 98
• catenary, 105
• cissoid, 95
• clothoid, 105
• concave, 228
• conchoid of Nicomedes, 96
• convex, 228
• corner point, 232
• cosecant, 76
• cosine, 75
• cotangent, 76
• curvature, 228
• cuspidal point, 232
• cycloid, 100
• --- damped oscillation, 83
• --- directing, 154
• --- double point, 232
• --- empirical, 106
• --- envelope, 236
• --- epicycloid, 101
• --- epitrochoid, 102
• --- equation
• complex form, 701
• --- plane, 194
4
• 225
3
• --- second degree, 205
• --- second order, 205
• --- space, 214
• --- error curve, 72
• --- evolute, 236
• --- evolvent, 236
4
• 237
3
• --- of the circle, 104
• --- exponential, 71
• --- fourth order, 96
• --- Gauss error curve, 759
• --- general discussion, 235
• --- hyperbolic cosine, 88
• --- hyperbolic cotangent, 89
• --- hyperbolic sine, d>^
• --- hyperbolic tangent, 88
• --- hyperbolic type, 68
4
• 70
3
• --- hyperbolice spiral, 104
• --- hypocycloid, 102
• --- hypotrochoid, 102
• --- imaginary, 194
• --- inflection point, 231
• --- involute, 237
• --- isolated point, 232
• --- Koch curve, 823
• --- lemniscate, 99
• --- length, line integral, first type, 464
• curve (continued II)
• --- logarithmic, 71
• --- logarithmic spiral, 104
• --- loops, 1007
• --- Lorentz curve, 94
• --- multiple point, 233
• --- normal, plane, 226
• --- 7j.-th degree, 63
4
• 194
3
• --- 7i.-th order, 63
4
• 194
3
• --- parabolic type, 63
• --- Pascal limacon, 96
• --- plane, 225
• --- angle, 228
• --- direction, 225
• --- vertex, 232
• --- quadratic, 205
• --- radius of curvature, 228
4
• 229
3
• --- representation with splines, 931
• --- secant, 76
• --- second degree, 205
• --- second order, 205
• --- semicubic parabola, 93
• --- sine, 75
• --- space, 238
• --- spherical, 158
4
• 172
• 643
3
• --- spiral, 103
• --- strophoide, 95
• --- tacnode, 232
• --- tangent, 76
• --- plane, 226
• --- terminal point, 232
• --- third degree, 65
• --- third order, 93
• --- tractrix, 106
• --- transcendent, 194
• --- trochoid, 100
• --- witch of Agnesi, 94
• curves
• --- family of, envelope, 237
• --- second order
• central curves, 205
• --- parabolic curves, 206
• --- polar equation, 207
• --- spherical, intersection point, 178
• cut, 292
• --- Dedekind's, 297
• --- fuzzy sets, 364
• --- set, 292
• cutting plane method, 877
• cycle, 351
• chain, 357
• --- limit, 833
• cycloid, 100
• --- basis, 100
• common, 100
• congruent, 100
• curtate, 100
• --- prolate, 100
• cylinder, 154
4
• 223
3
• circular, 155
• --- elliptic, 223
• --- hollow, 155
• --- hyperbolic, 223
• cylinder (continued)
• --- invariant signs
• --- elliptic, 224
• --- hyperbolic, 224
• --- parabolic, 224
• --- parabolic. 223
• cylindrical coordinates, 209
• cylindrical function, 509
• cylindrical surface, 154
4
• 213
3
2
1
• D
2
• D
• d'Alembert formula, --- flow,
3
• d'Alembert formula, 53
• damping parameter, 897
• damping, oscillations, 83
• Darboux vector, 243
• data type, 340
• death process, 769
• debt, 23
• decay, radioactive, 768
• decimal
• --- number, 937
• --- normalized, 939
• --- representation, 937
• --- system, 937
• decoding, 331
• decomposition
• --- orthogonal, 616
• --- partial fractions, 15
• --- vectors, 182
• decomposition theorem
• differential equation, higher order, 501
• Dedekindcut, 297
• defect, 909
4
• 913
3
• --- vector space, 318
• definite, positive, 893
• defuzzification, 373
• degeneracy of states, 541
• degree
• --- curve, second degree, 205
• --- curve, n-th degree, 194
• --- homogeneity, 120
• --- in-degree, 348
• --- matrix, 355
• --- measure in degrees, 130
• --- out-degree, 348
• Delambre equations, 166
• --- S distribution, 640
• --- S function, 636
4
• 640
3
• --- application, 717
• --- approximation, 717
• Dirac, 714
• --- S functional, 623
• De Morgan
• --- law. 293
• --- rule, 287
• --- Boolean algebra, 343
• density function, 752
• --- multidimensional, 754
• dependence, linear, 272
4
• 317
3
• deposit
• --- in the course of the year, 22
• --- regular, 22
• --- single, 22
• depreciation, 26
• --- arithmetically declining, 26
• digital, 27
• --- geometrically declining, 27
• --- straight-line, 26
• derivative
• --- complex function, 671
• --- constant, 380
• directional, 649
• --- scalar field, 649
• --- vector field, 650
• distribution, 641
• --- exterior, 382
• --- fraction, 382
• --- Frechet, 632
• --- function
• --- composite, 382
• --- elementary, 380
• --- implicit, 383
• --- inverse, 383
• --- parametric representation, 384
• --- several variable, 392
• --- generalized, 640
• --- higher order, 385
4
• 395
3
• --- inverse function, 387
• --- parametric representation, 387
• --- interior, 382
• --- left-hand, 380
• --- logarithmic, 382
• --- mixed, 395
• --- one variable, 379
• --- partial, 392
• --- product, 381
• --- quotient, 382
• --- right-hand, 380
• --- scalar multiple, 380
• --- Sobolev sense, 640
• --- space, 649
• --- sum, 380
• --- table, 381
• --- vector function, 642
• --- volume differentiation, 650
• Derive (computer algebra system), 953
• Descartes rule, 45
• descent, 898
• descent method, 868
• determinant, 259
• differentiation, 261
• --- evaluation, 261
• --- functional, 121
• --- Jacobian, 121
• --- multiplication, 260
• --- reflection, 260
• --- rules of calculation, 260
• --- Wronskian, 500
4
• 802
3
• --- zero value, 260
• determination of extrema
• --- absolute extremum, 392
• --- implicit function, 392
• deviation, standard deviation, 753
• devil's staircase, 844
• diagonal matrix, 252
• diagonal method, Maxwell, 684
• diagonal strategy, 892
• diameter
• --- parabola, 204
• --- circle, 139
• --- conjugate
• --- ellipse, 199
• --- hyperbola, 202
• --- ellipse, 199
• --- hyperbola, 202
• diffeomorphism, 798
• --- Anosov, 829
• --- orientation-preserving, 844
• difference
• --- bounded, 366
• --- finite expression. 908
• --- sets, 293
• --- symmetric, 293
• --- significant, 779
• --- symmetric
• --- sets, 293
• --- z-transformation, 735
• difference equation, 733
• --- boundary value, 740
• --- linear, 738
• --- partial differential equations, 911
• --- second-order, 739
4
• 740
3
• difference method, 908
• --- partial differential equations, 911
• difference quotient, 898
• difference schema, 18
• differentiability
• --- complex function, 671
• --- function of one variable, 379
• --- function of several variables, 394
• --- with respect to the initial conditions, 797
• differentiable
• --- continuously, 632
• --- Frechet, 632
• differential
• --- arc
• --- plane, 226
• --- surface, 245
• --- complete, 394
• --- first-order, 380
• --- higher order, 394
• --- integrability, 468
• --- notion, 393
• --- partial, 394
• --- principal properties, 394
• --- quotient (see also derivative), 379
• --- second-order. 396
• --- total, 394
4
• 395
3
• --- n-th order, 396
• --- second-order, 396
• differential calculus, 379
• --- fundamental theorem, 388
• differential equation, 487
• --- boundary value problem, 514
• --- eigenfunction, 515
• --- eigenvalue, 515
• --- eigenvalues, 541
• --- flow, 797
• --- Fourier transformation, 731
• --- Laplace transformation, 721
• --- numerical solution, 497
• --- operational notation, 502
• --- order, 487
• --- orthogonality, 515
• --- self-adjoint, 514
• --- singular solution. 489
• --- stiff, 908
• --- topological equivalent, 810
• --- Weber, 545
• differential equation, higher order, 497
• --- constant coefficients, 500
• decomposition theorem, 501
• --- Euler, n-th order, 504
• --- fundamental system, 500
• --- n-th order, 500
• --- lowering the order, 499
4
• 501
3
• --- normal form, 505
• --- quadrature, 501
• --- superposition principle, 501
• --- system, 497
• --- system of linear, 505
• --- system of solutions, 498
• --- variation of constants, 501
• differential equation, linear
• --- autonomous, on the torus, 803
• --- first-order, 801
• --- fundamental theorem, 801
• --- homogeneous, 801
• --- homogeneous systems, 506
• --- inhomogeneous, 801
• --- inhomogeneous systems, 506
• --- matrix-differential equation, 801
• --- non-autonomous, on the torus, 844
• --- periodic coefficients, 803
• --- second order, 507
• --- Bessel, 509
• defining equation, 508
• --- Hermite, 514
• --- hypergeometric, 513
• --- Laguerre, 513
• --- Legendre, 511
• --- method of unknown coefficients, 507
• differential equation, ordinary, 487
• --- approximate integration, 904
• --- autonomous, 800
• --- Bernoulli, 491
• --- boundary value conditions, 487
• --- boundary value problem, 487
• --- center, 495
• --- central point, 495
• --- Clairaut, 493
• direction field, 488
• --- element, singular, 493
• --- exact, 489
• --- existence theorem, 488
• --- explicit, 487
• --- first-order, 488
• --- approximation methods, 496
• --- important solution methods, 489
• --- flow, 797
2
• fraction of linear functions, --- division
3
• --- fraction of linear functions, 494
• --- general solution, 487
• --- general, n-th order, 487
• differential equation, ordinary (continued)
• --- graphical solution, 497
• --- homogeneous, 489
• --- implicit, 487
4
• 492
3
• --- initial value conditions, 487
• --- initial value problem, 487
• --- integral, 487
• --- integral curve, 488
• --- integral, singular, 493
• --- integrating factor, 490
• --- Lagrange, 492
• --- linear, 731
• --- constant coefficients, 721
• --- first-order, 490
• --- variable coefficient, 722
• --- linear, planar, 797
4
• 811
3
• --- multiplier, 490
• --- non-autonomous, 800
• --- notion, 487
• --- particular solution, 487
• --- point, singular, 493
• --- radicals, 493
• --- ratio of arbitrary functions, 496
• --- Riccati, 491
• --- separation of variables, 489
• --- series expansion, 496
• --- singular point, 494
• --- solution, 487
• --- successive approximation, 496
• --- van der Pol, 833
• differential equation, partial, 517
• --- approximate integration, 911
• --- Cauchy-Riemann, 530
4
• 672
3
• --- characteristic system, 518
• --- Clairaut, 520
• --- completely integrable, 521
• --- eigenfunction, 541
• --- electric circuit, 531
• --- elliptic type, 522
• --- constant coefficients, 524
• --- Euler, calculus of variations, 554
• --- field theory, 669
• --- first-order, 517
• --- canonical system, 519
• --- characteristic strip, 519
• --- characteristic system, 517
• --- linear, 517
• --- non-linear, 519
• --- non-linear, complete integral, 519
• --- total differentials, 521
• --- two variables, 520
• --- Fourier transformation, 732
• --- Hamilton, 801
4
• 841
3
• --- heat conduction equation
• --- one dimensional, 531
• --- one-dimensional, 529
• --- three-dimensional, 537
• --- Helmholtz, 540
• --- hyperbolic type, 522
4
• 524
3
• --- integral surface, 518
• --- Laplace, 669
• --- Laplace transformation, 723
• --- linear, 517
• differential equation, partial (continued)
• --- non-linear, 546
• --- Schrodinger, 549
• --- normal system, 519
• --- notion, 487
• --- parabolic type, 522
• --- constant coefficients, 524
• --- Poisson, 538
4
• 667
• 670
3
• --- quasilinear, 517
• --- reduced, 830
• --- reduced, normal form, 830
• --- second-order, 522
• --- constant coefficients, 524
• --- separation of variables, 540
• --- ultrahyperbolic type, constant coefficients, 524
• differential operation
• --- review, 658
• --- vector components, 659
• differential operator
• divergence, 653
4
• 658
3
• --- gradient, 650
4
• 651
• 658
3
• --- Laplace, 657
4
• 658
3
• --- nabla, 656
4
• 658
3
• --- non-linear, 632
• --- relations, 658
• --- rotation, 654
4
• 658
3
• --- rules of calculations, 658
• --- space, 649
4
• 656
3
• --- vector gradient, 652
• differential transformation, affine, 674
• differentiation, 379
• --- complex function, 671
• --- composite function, 397
• --- function
• --- elementary, 380
• --- implicit, 383
• --- inverse, 383
• --- one variable, 379
• --- parametric representation, 384
• --- several variables, 392
• --- graphical, 384
• --- higher order
• --- inverse function, 387
• --- parametric representation, 387
• --- implicit function, 398
• --- logarithmic, 382
• --- several variables, 397
• --- under the integration sign, 459
• --- volume, 650
• differentiation rules
• --- basic rules, 380
• derivative of higher order, 385
• --- function
• --- one variable, 380
• --- several variables. 380
4
• 395
3
• --- table, 386
• --- vector function, 642
• --- vectors, 642
• diffusion coefficient, 538
• diffusion equation, 551
• --- three-dimensional, 537
• digon, spherical, 161
• dihedral angle, 150
• dihedral group, 299
• dimension, 822
• --- capacity, 823
• --- correlation, 825
• defined by invariant measures, 824
• Douady-Oesterle, 826
• --- formula, 317
• --- generalized. 825
• --- Hausdorff, 822
• --- information, 824
• --- lower pointwise, 824
• --- Lyapunov, 825
• --- measure, 824
• --- metric, 822
• --- Renyi, 825
• --- upper pointwise, 824
• --- vector space, 317
4
• 599
3
• Dirac
• distribution, 640
• --- measure, 636
4
• 816
3
• --- theorem, 353
• directing curve, 154
• direction
• --- cosine, space, 210
• --- plane curve, 225
• --- space, 210
• --- space curve, 238
• directional
• --- angle, 145
• derivative, 651
• directrix
• --- ellipse, 198
• --- hyperbola, 201
• --- parabola, 203
• Dirichlet
• --- condition, 422
• --- problem, 528
4
• 670
3
• discontinuity, 57
• --- function, 57
• --- removable, 58
• discount, 21
• discretization error
• --- global, 907
• --- local, 907
• discretization step interval, 898
• discriminant, 522
• disjoint, 292
• disjunction, 286
• --- elementary, 345
• dispersion, 753
• dissolving torus, 841
• distance, 451
• --- Hamming distance, 604
• --- line-point, 195
• --- metric space, 604
• --- planes, 217
• --- parallel, 217
• --- point-line, space, 218
• --- point-plane, space, 215
• --- spherical, 158
• --- two lines, space, 218
• --- two points, 191
• --- space, 212
• distribution, 639
4
• 640
• 717
3
• --- binomial, 755
• --- X2, 762
• --- continuous, 758
• derivative, 640
• Dirac, 640
• discrete, 754
• --- exponential, 760
• --- Fisher, 763
• --- frequency, 772
• --- function, 751
• --- function, continuous, 752
• --- hypergeometric, 755
4
• 756
3
• --- logarithmic normal, 759
• --- lognormal distribution, 759
• --- measurement error density, 788
• --- normal, 758
• --- Poisson, 757
• --- regular, 640
• --- standard normal, 759
• --- Student, 763
• --- t distribution, 763
• --- theory. 714
• --- Weibull, 761
• distribution problem, 860
• distributive law
• --- Boolean algebra, 342
• --- matrices, 254
• --- propositional logic, 287
• --- ring, field, 313
• --- sets, 293
• --- tensors, 264
• distributivity, product of vectors, 184
• divergence, 667
• --- central field, 654
• definition, 653
• different coordinates, 653
• --- improper, 405
• --- proper, 405
• --- remark, 650
• --- sequence of numbers, 405
• --- series, 409
• --- theorem, 665
• --- vector components, 659
• --- vector field, 653
• divisibility, 320
• --- criteria, 322
4
• 323
3
• division
2
• complex numbers, --- Duhamel formula,
3
• --- complex numbers, 37
• --- computer calculation, 940
• --- external, 192
• --- golden section, 193
• --- harmonic, 192
• --- in extreme and mean ratio, 193
• --- internal, 192
• --- line segment, plane, 192
• --- polynomial, 14
• --- rational numbers, 1
• --- segment, space, 212
• divisor, 320
• --- greatest common (g.c.d.)
• --- integer numbers, 323
• --- linear combination, 324
• divisor (continued)
• --- polynomials, 14
• --- positive, 322
• dodecahedron, 154
• domain, 118
• --- closed, 118
• --- convergence, function series, 414
• doubly-connected, 118
• --- function, 47
• --- image, set, 48
• --- multiply-connected, 118
• --- non-connected, 118
• --- of attraction, 800
• --- of individuals, predicate calculus, 289
• --- open, 118
• --- operator, 600
• --- set, 48
• --- simply-connected, 118
4
• 687
3
• --- three or multidimensional, 118
• --- two-dimensional, 118
• --- values, set, 48
• dot product, 183
• double integral, 471
• --- application, 474
• --- notion, 471
• double line, 205
• dual, 343
• duality
• --- linear programming, 856
• --- non-linear optimization, 863
• --- theorem, strong, 864
• duality principle, Boolean algebra, 343
• dualizing, 343
• Duhamel formula, 722
2
1
• E
2
• E
• eccentricity, numerical --- vector space,
3
• eccentricity, numerical
• --- curve second order, 207
• ellipse, 198
• --- hyperbola, 200
• --- parabola, 203
• edge
• --- angle, 150
• --- figure, 150
• --- graph, 348
• --- length, 350
• --- valuation, 350
• --- multiple, 348
• --- sequence, 351
• --- cycle, 351
• --- directed circuit, 351
• --- isolated edge, 351
• --- open, 351
• --- path, 351
• effective rate, 22
• eigenfunction, 541
• --- differential equation, 515
• --- integral equation, 565
4
• 570
3
• --- normalized, 516
• eigenvalue, 278
4
• 541
3
• --- differential equation, 515
• --- integral equation, 565
4
• 570
3
• --- operator, 622
• eigenvalue problem
• --- general, 278
• --- matrices, 278
• --- special, 278
• eigenvector, 264
4
• 278
• 622
3
• Einstein's summation convention, 262
• element, 290
• --- finite, 914
• --- generic, 814
• --- linearly independent, 598
• --- neutral, 298
• --- positive, 601
• element of area, plane, table, 474
• element of surface, curved, table, 482
• element of volume, table, 479
• elementary cell
• --- crystal lattice, 310
• --- non-primitive, 310
• --- primitive, 310
• elementary formula, predicate logic, 289
• elementary surface, parametric form, 482
• elementary volume
• --- arbitrary coordinates, 479
• --- Cartesian coordinates, 476
• --- cylindrical coordinates, 477
• --- spherical coordinates, 478
• elimination method, Gauss, 276
• elimination step, 891
• ellipse, 198
• --- arc, 200
• --- area, 199
• --- diameter, 198
• equation, 198
• --- focal properties, 198
• --- focus, 198
• --- perimeter, 200
• --- radius of curvature, 199
• --- semifocal chord, 198
• --- tangent, 199
• --- transformation, 205
• --- vertex, 198
• ellipsoid, 220
• --- central surface, 224
• --- cigar form, 220
• --- imaginary, 224
• --- lens form, 220
• --- of revolution, 220
• --- surface second order, 224
• embedding, canonical, 626
• encoding method, RSA, 331
• encyphering, 334
• endomorphism, linear operators, 601
• endpoint, 348
• energy
• --- particle, 539
• --- spectrum, 540
• --- system, 538
• --- zero-point translational energy, 541
• --- zero-point vibration energy, 546
• entropy
• --- generalized, 825
• --- metric, 819
• --- topological, 819
4
• 829
3
• envelope, 236
4
• 237
3
• epicycloid, 101
• --- curtate, 102
• --- prolate, 102
• epitrochoid, 102
• equality
• --- asymptotic, 419
• --- complex numbers, 34
• --- matrices, 254
• equality relation
• --- identity, 10
• equation, 10
• --- algebraic, 38
4
• 43
3
• --- biquadratic, 42
• --- characteristic, 278
4
• 494
• 503
• 505
3
• --- cubic, 40
4
• 63
3
• --- curve
• --- plane, 194
4
• 225
3
• --- second degree, 205
• --- second order, 205
• --- defining, 508
• --- degree, 38
• --- Diophantine, 325
• --- linear, 325
• ellipse, 198
• exponential, solution, 45
• --- first degree, 39
• --- fourth degree, 42
• --- homogeneous, 629
• --- hyperbola, 200
• --- hyperbolic functions, solution, 46
• --- inhomogeneous, 629
• --- irrational, 39
• --- Korteweg de Vries, 548
• --- line
• --- plane, 194
• --- space, 217
• --- linear, 39
• --- logarithmic, solution, 46
• --- logistic, 798
4
• 837
3
• --- non-linear
• --- fixed point, 884
• --- numerical solution, 884
• --- normal form, 38
• --- n-th degree, 43
• --- operator equation, 629
• --- parabola, 203
• --- Parseval, 422
4
• 516
• 579
• 618
3
• --- pendulum, 841
• --- plane, 214
• --- general, 214
• --- Hessian normal form, 214
• --- intercept form, 215
• --- plane curve, 194
• --- polynomial
• --- numerical solution, 887
• --- quadratic, 39
4
• 62
3
• --- root, definition, 38
• --- Schrodinger
• --- linear, 538
• --- non-linear, 547
• --- second degree, 39
• --- sine-Gordon, 549
• --- solution, general, 38
• --- space curve, 214
4
• 238
3
• --- vector form, 238
• --- sphere, 243
• --- surface
• --- normal form, 220
• --- second order, 223
• --- space, 213
4
• 243
3
• --- system of, 38
• --- term algebra, 341
• --- third degree, 40
• --- transcendent, 38
• --- trigonometric, solution, 46
• --- vector, 187
• equation system
• --- linear^ 271
4
• 890
3
• --- overdetermined, 893
• --- non-linear, 890
4
• 896
3
• --- numerical method, iteration, 890
• --- numerical solution, 890
• --- direct method, 890
• --- overdetermined, 277
4
• 890
3
• --- row echelon form, 890
• --- underdetermined, 890
• equations
• --- Delambre, 166
• --- L'Huilier. 167
• --- Mollweide, 142
• --- Neper, 167
• equilibrium point, 797
• --- hyperbolic, 804
• equivalence, 286
• --- class, 297
• --- logic, 287
• --- proof, 5
• --- relation, 296
• Eratosthenes sieve, 320
• error
• --- absolute, 792
4
• 939
3
• --- absolute maximum error, 792
• --- apparent, 790
• --- average, 789
4
• 791
3
• --- computer calculation, 941
• --- defined, 792
• --- density function, 788
• --- discretization, 942
• equation, 893
• estimation, iteration method, 897
• --- input error, 941
• --- least mean squares, 421
• --- mean, 789
• --- square, 790
4
• 791
3
• --- normally distributed, 788
• --- percentage, 792
• --- probable, 789
4
• 791
3
• --- relations between error types, 789
• --- relative, 792
4
• 939
3
• --- relative maximum error, 792
• --- round-off, 942
• --- single measurement, 790
• --- standard, 790
• --- true, 790
• --- truncation error, 942
• --- type 1
4
• 753
3
• error (continued)
• --- type, measurement errors. 788
• error analysis, 796
• error calculus
• --- direct problem, 941
• --- inverse problem, 941
• error curve, 72
• error estimation, mean value theorem, 389
• error function, 461
• --- Gauss, 759
• error integral, Gauss, 461
• error orthogonality, 909
• error propagation. 794
• --- law, 795
• error types, computer calculation, 941
• ess. sup (essential supremum), 639
• Euclidean
• --- algorithm, 3
4
• 14
3
• --- polynomial rings, 315
• --- norm, 318
• --- vector norm, 258
• --- vector space, 318
2
• Euler --- relative,
3
• Euler
• --- angle, 211
• --- broken line method, 904
• --- circuit, 352
• --- constant, 460
• --- differential equation
• --- n-th order, 504
• --- variational calculus. 554
• --- formula, 247
4
• 420
3
• --- formulas, 421
4
• 424
3
• --- function, 331
• --- graph, 352
• --- integral of the second kind, 459
• --- integral, first kind, 1056
• --- integral, second kind, 1056
• --- numbers, 412
• --- polygonal method, 904
• --- relation (complex numbers), 35
4
• 698
3
• --- theorem, 154
• --- trail, 352
• --- open, 352
• Euler-Hierholzer theorem, 352
• event, 747
• --- certain, 747
• --- complete system of, 748
4
• 750
3
• elementary, 747
• --- impossible, 747
• --- independent, 750
• --- random, 747
• --- set of events, 747
• --- simple, 747
• evolute, 236
• evolution
• equation, 547
• --- function, 547
• evolution strategies, 870
• evolvent, 236
4
• 237
3
• --- of circle, 104
• excess, spherical triangle, 163
• exchange theorem, 395
• exchange, cyclic, sides and angles, 141
• excluded middle, 287
• existential quantifier, 289
• expansion
• --- Fourier, forms, 423
• --- Laplace, 259
• --- Laurent, 692
• --- Maclaurin, 418
• --- Taylor, 389
4
• 417
• 691
3
• expectation, 753
• expected value, 753
• --- bivariate distribution, 779
• --- two-dimensional distribution, 779
• exponent, 7
• exponential distribution, 760
• exponential function, 71
• --- complex, 679
• --- general, 699
• --- natural, 698
• exponential integral, 461
• exponential sum, 72
• expression
• --- algebraic, 10
• --- manipulation, 979
• --- analytic, 48
• --- domain, 48
• explicit form, 48
• --- implicit form, 48
• --- parametric form, 49
• --- Boolean, 344
• --- concurrent, 345
• explicit form, 48
• --- finite, 908
• --- partial differential equations, 911
• --- implicit form, 48
• --- integral rational, 11
• --- irrational, 11
4
• 17
3
• --- non-polynomial, manipulation, 981
• --- parametric form, 49
• --- propositional logic, 286
• --- rational, 11
4
• 14
3
• --- semantically equivalent, 345
• --- tautology, 288
• --- transcendent, 11
• --- vector analysis, 659
• extension principle, 368
• extension theorem of Hahn, 625
• extension, linear functional, 624
• extraction of the root
• --- complex numbers, 38
• --- real numbers, 8
• extrapolation principle, 902
• extremal, radius of curvature, 558
• extreme value, 390
• --- absolute, 390
• --- determination, 391
• --- higher derivatives, 391
• --- side conditions, 403
• --- sign change, 390
• --- function, 401
• --- relative, 390
2
1
• F
2
• F
• face, corner, --- discrete,
3
• face, corner, 151
• factor algebra, 340
• factor group, 302
• factor ring, 314
• factor, polynomial, product representation, 43
• factorial, 13
• --- generalization of the notion, 462
• factoring out, 11
• Falk scheme, 254
• feasible set, 847
• Feigenbaum constant, 837
4
• 839
3
• FEM (finite element method), 913
• FFT (fast Fourier transformation), 928
• Fibonacci
• --- numbers, 325
4
• 845
3
• --- sequence, 325
• field (algebraically), 313
• --- extension, 313
• finite, 314
• --- Galois, 314
• field
• --- axial, 644
• --- central symmetric, 643
• --- circular, 646
• --- conservative, 662
• --- Coulomb, point-like charge, 646
4
• 668
3
• --- cylinder symmetric. 644
• flow, 664
• function, 681
• --- gravitational, point mass, 669
• --- Newton, point-like mass, 646
• --- potential, 662
• --- scalar, 643
• --- source, 667
• --- spherical, 643
• field theory
• --- basic notions, 642
• --- differential equations, partial, 669
• fields, superposition, 669
• finite difference method, 534
• finite element method, 534
4
• 913
3
• fitting problem
• --- different versions, 403
• --- linear, 893
• --- non-linear, 107
• fixed point
• --- conformal mapping, 675
4
• 676
3
• flip bifurcation, 836
• fixed-point number, 938
• floating-point number, 938
• --- IEEE standard, 939
• --- Maple, 970
• --- Mathematica, 957
• --- semilogarithmic form, 938
• Floquet
• --- representation, 805
• --- theorem, 803
• flow
• --- differential equation, 797
• --- edge, 358
• --- scalar field, 664
• --- vector field
• --- scalar flow, 664
• --- vector flow, 664
• focal line, tractrix, 106
• focus, 809
• --- compound, 832
• --- ellipse, 198
• --- hyperbola, 200
• --- parabola, 203
• --- saddle, 804
• --- saddle focus, 809
• --- stable, 804
• form
• --- quadratic, 893
• --- saddle, 249
• formula
• --- binomial, 12
• --- Cardano, 41
• --- closed, predicate logic, 289
• --- d'Alembert, 536
• --- Duhamel, 722
• --- Euler, 247
4
• 420
3
• --- Heron's, 143
• --- integral (series remainder), 418
• --- interpretation, predicate logic, 289
• --- Kirchhoff, 536
• --- Lagrange, 418
• --- Leibniz, 385
• --- Liouville, 501
4
• 802
3
• --- Maclaurin, 415
• --- manipulation, 953
• --- de Moivre
• --- complex number, 37
• --- hyperbolic functions, 90
• --- trigonometric functions, 79
• --- Parseval. 729
• --- Pesin, 821
4
• 828
3
• --- Plemelj and Sochozki, 593
• --- Poisson, 536
• --- predicate logic, 289
• --- rectangular, 899
• --- Riemann, 530
• --- Simpson, 900
• --- Stirling, 462
• --- tangent, 142
• --- tautology, 290
• --- Taylor
• --- one variable, 389
• --- several variables, 397
• --- trapezoidal, 899
• formulas
• --- Euler, 421
4
• 424
3
• Frenet, 243
• --- predicate logic, 289
• four-group, Klein's, 302
• Fourier analysis, 420
• Fourier coefficient, 420
4
• 927
3
• --- asymptotic behavior, 422
• --- determination, 403
• --- numerical methods, 424
• Fourier expansion, 420
• --- complex functions, 426
• forms, 423
• --- symmetries, 422
• Fourier integral, 424
4
• 724
3
• --- complex representation, 724
• --- equivalent representations, 724
• Fourier series, 421
• --- best approximation, 617
• --- complex representation, 421
• --- Hilbert space, 617
• --- Bessel inequality, 618
• Fourier sum, 421
4
• 927
3
• --- complex representation, 928
• Fourier transformation, 724
• --- addition law, 727
• --- comparing to Laplace transformation, 730
• --- convolution, 729
• --- two-sided, 713
• --- definition, 725
• --- differentiation
• --- image space, 728
• --- original space, 728
• --- discrete complex, 928
• fast, 928
• Fourier cosine transformation, 725
• Fourier exonential transformation, 726
• Fourier sine transformation, 725
• frequency-shift theorem, 728
• --- integration
• --- image space, 728
• --- original space, 729
• --- inverse, 725
• --- linearity law, 727
• --- shifting theorem, 728
• --- similarity law, 728
• --- spectral interpretation, 726
• --- survey, 707
• --- tables, 726
• --- transform, 730
• fractal, 822
• fractile, 752
• fraction
• --- continued, 3
• --- decimal, 1
• --- improper, 15
• --- proper, 15
• fractional part of x, 49
• frames, 743
• Frechet
• --- derivative, 632
• --- differential, 632
• Fredholm
• --- alternative, 629
• --- alternative theorem, 571
• --- integral equation, 563
4
• 609
3
• first kind, 577
• --- solution method, 569
• --- theorems, 569
• Frenet formulas, 243
• frequency, 75
• --- angular/radial, 82
• --- distribution, 772
• --- locking, 844
• --- relative, 748
• --- sine, 82
• --- spectrum, 727
• --- continuous, 425
• --- discrete, 425
• --- statistics, 748
4
• 772
3
• --- cumulative, 773
• Fresnel integral, 697
• frustum
• --- cone, 156
• --- pyramid, 152
• function, 47
• --- absolutely integrable, 455
4
• 458
3
• --- algebraic, 60
• --- amplitude function, elliptic, 703
• --- analytic, 672
• --- area, 91
• --- arrow, 294
• --- autocorrelation, 818
• --- Bessel, 509
• --- modified, 509
• --- beta function, 1056
• --- Boolean, 287
4
• 344
3
• --- bounded, 50
• --- circular, geometric definition, 130
• --- comparable, 558
• --- complement, 368
• --- complex, 47
4
• 671
3
• --- algebraic, 698
• --- bounded, 673
• --- linear, 675
• --- linear fractional, 676
• --- quadratic, 677
• --- square root, 677
• --- complex variable, 671
• --- complex-valued, 47
• --- composite, 62
• --- derivative, 382
• --- intermediate variable, 382
• --- continuity
• --- in interval, 57
• --- one-sided, 57
• --- piecewise, 57
• --- continuous, complex, 671
• --- cosecant, 76
• --- cosine, 75
• --- cotangent, 76
• --- cyclometric, 84
• --- cylindrical, 509
• --- density, 752
• --- dependent, 121
• --- discontinuity, 57
• --- removable, 58
• --- discrete, 734
2
• distribution, --- union,
3
• --- distribution, 751
• --- double periodic, 704
• --- elementary, 60
• --- elementary, transcendental. 698
• --- elliptic, 437
4
• 693
• 704
3
• --- entire rational, 60
• --- error function, 461
• --- Euler, 331
• --- even, 50
• --- exponential, 61
4
• 71
3
• --- exponential, complex, 679
• --- exponential, natural, 698
• function series, 414
• --- gamma, 461
• --- generalized, 639
4
• 640
• 717
3
• function (continued I)
• --- Green, 532
• --- Hamilton, 520
4
• 801
3
• --- harmonic. 669
4
• 672
3
• --- Heaviside, 641
• --- Hermite, 616
• --- holomorphic, 672
• --- homogeneous, 120
• --- homographic, 61
4
• 65
3
• --- hyperbolic, 87
4
• 699
3
• --- geometric definition, 130
4
• 131
3
• --- impulse, 714
• --- increment, 907
• --- independent, 121
• --- integrable, 441
• --- integral rational, 60
• first degree, 62
• --- n-th degree, 63
• --- second degree, 62
• --- third degree, 63
• --- inverse, 51
• --- complex, hyperbolic, 699
• --- complex, trigonometric, 699
• --- derivative, 383
• --- derivative of higher order, 387
• --- existence, 60
• --- hyperbolic, 91
• --- trigonometric, 61
4
• 84
3
• --- irrational, 61
4
• 69
3
• --- Jacobian, 703
• --- Lagrangian, 862
• --- Laguerre, 616
• --- Laplace, 669
• --- limit, 51
• --- at infinity, 53
• --- infinity, 52
• --- iterated, 122
• --- left-hand, 52
• --- right-hand, 52
• --- Taylor expansion, 55
• --- limit theorems, 53
• --- linear, 60
4
• 62
3
• --- linear fractional, 61
4
• 65
3
• --- local summable, 639
• --- logarithm, complex, 678
• --- logarithmic, 61
4
• 71
3
• --- loop function, 1007
• --- MacDonald, 509
• --- matrix exponential, 802
• --- mean value, 444
• --- measurable, 636
• --- meromorphic, 693
4
• 704
• 719
3
• --- monotone
• --- decreasing, 49
• --- increasing, 49
• --- non-elementary, 60
• --- notion, 47
• --- odd, 50
• --- of angle, 74
• --- one variable, 47
• --- order of magnitude, 55
• --- parametric representation
• --- derivative of higher order, 387
• function (continued II)
• --- periodic, 50
4
• 716
3
• --- piecewise continuous, 57
• --- point of discontinuity, 57
• finite jump, 58
• --- tending to infinity, 57
• --- positive homogeneous, 558
• --- power, 70
• --- primitive, 427
• --- quadratic, 60
• --- random variable, 751
• --- rational, 61
4
• 64
3
• --- real, 47
• --- regular, 672
• --- Riemann, 530
• --- sample function, 770
• --- secant, 76
• --- several variables, 47
4
• 117
3
• --- sign of, 49
• --- sine, 74
• --- special fractional linear fractional, 64
• --- step, 734
• --- stream, 681
• --- strictly monotone, 49
• --- sum of linear and fractional linear functions, 678
• --- summable, 637
• --- switch, 346
• --- tangent, 76
• --- theory, 671
• --- theta, 705
• --- transcendental, 61
• --- trigonometric, 61
4
• 74
• 699
3
• --- geometric definition, 130
• --- truth, 286
4
• 287
3
• --- Weber, 509
• --- Weierstrass, 705
• function system
• --- orthogonal, 920
• --- orthonormal, 920
• function theory, 671
• functional, 553
4
• 619
3
• --- definition, 47
• --- linear, 319
4
• 600
• 601
3
• --- linear continuous, 623
• --- L space, 624
• functional determinant, 121
4
• 266
• 474
3
• fundamental form
• first quadratic, of a surface, 245
• --- second quadratic, of a surface, 248
• fundamental formulas
• --- spherical trigonometry, 164
• fundamental laws, set algebra, 292
• fundamental matrix, 802-804
• fundamental problem
• first, triangulation, 147
• --- second, triangulation, 148
• fundamental system
• --- differential equation, higher order, 500
• fundamental theorem
• --- Abelian groups, 302
• --- algebra, 43
• --- elementary number theory, 321
• --- integral calculus, 442
• future value, 25
• fuzzy
• --- control, 374
• --- inference, 372
• --- linguistics, 361
• --- logic, 360
• --- logical inferences, 372
• --- product relation, 371
• --- relation, 369
• --- relation matrix, 370
• --- system, 377
• --- systems, applications, 374
• --- valuation, 369
• fuzzy set
• --- aggregation, 365
• --- aggregation operator, 370
• --- complement, 365
• --- composition, 371
• --- cut (representation theorem), 364
• --- degree, 364
• --- empty, 363
• --- intersection, 365
• --- intersection set, 366
• --- level set, 364
• --- normal, 364
• --- peak, 363
• --- similarity, 364
• --- subnormal, 364
• --- subset, 363
• --- support, 360
• --- tolerance interval, 363
• --- union, 365
• --- universal, 363
• fuzzy sets
• --- cut, 364
• --- intersection, 366
• --- union, 366
2
1
• G
2
• G
• g.c.d. (greatest common divisor), 324
• g.c.d. and l.c.m., relation between, 324
• Gab or transformation, 743
• Galerkin method. 909
• Galois field, 314
• gamma function, 459
3
• 461
2
• Gauss
• --- algorithm, 276
3
• 891
2
• --- coordinates, 244
• --- curvature, surface, 249
• --- elimination method, 276
3
• 890
2
• --- error curve, 759
• --- error function, 759
• --- error integral, 461
• --- error propagation law, 795
• --- integral theorem, 665
• --- least squares method, 403
3
• 894
• 919
2
• --- plane, 34
• --- step, 276
• --- transformation, 277
3
• 783
• 894
2
• Gauss-Kruger coordinates, 161
• Gauss-Newton method, 897
• --- derivative free, 898
• Gauss-Seidel method, 895
• generating line, 154
• generator, 154
• --- ruled surface, 222
• geodesic line, 250
• geodesy
• --- angle, 145
• --- coordinates, 143
• --- polar coordinates, 143
• geometric sequence, 19
• geometry, 128
• --- analytical, 180
• --- plane, 189
• --- space, 207
• --- differential, 225
• --- plane, 128
• Girard theorem, 164
• golden section, 2
3
• 4
• 193
• 845
2
• gon, 145
• gradient
• --- definition, 651
• --- different coordinates, 651
• --- remark, 650
• --- scalar field, 650
3
• 651
2
• --- vector components, 659
• --- vector gradient, 652
3
• 657
2
• Graeffe method, 889
• graph
• --- alternating way, 356
• --- arc, 348
• --- bipartite, 349
• --- complete, 349
• --- complete bipartite, 349
• --- components, 351
• --- connected, 351
3
• 357
2
• --- cycle, 357
• --- directed, 348
• --- directed circuit, 357
• --- directed edge, 348
• --- edge, 348
• --- Euler, 352
• --- flow, 358
• --- increasing way, 356
• --- infinite, 349
• --- isomorphism, 349
• --- loop, 348
• --- mixed, 348
• --- non-planar, 357
• --- partial, 349
• --- planar, 357
• --- plane, 349
• --- regular, 349
• --- simple, 348
• --- special classes, 349
• --- strongly connected, 357
• --- subdivision, 357
• --- subgraph, 349
• --- transport network, 349
• --- tree, 349
• --- undirected, 348
• --- vertex, 348
• --- weighted, 350
• graph paper
• --- double logarithmic, 115
• --- log-log paper, 115
• --- notion, 115
• --- reciprocal scale, 116
• --- semilogarithmic, 115
• graph theory, algorithm, 348
• gravitational field, point mass, 669
• great circle, 158
3
• 173
2
• greatest common divisor (g.c.d.)
• --- integer numbers, 323
• --- linear combination, 324
• --- polynomials, 14
• Green
• --- function, 532
• --- integral theorem, 666
• --- method, 531
3
• 532
2
• group, 299
• --- Abelian, 299
• --- dihedral, 299
• --- element, 299
• --- character, 304
• --- inverse, 299
• --- factor group, 302
• --- permutation, 301
• --- point group, 308
• --- representation, 303
• --- irreducible, 305
• --- reducible, 305
• --- subgroup,300
• --- symmetry group, 308
• group table, 300
• --- Cayley table, 300
• grouping, 11
• groups
• --- applications, 307
• --- direct product, 301
• --- homomorphism, 302
• --- homomorphism theorem, 302
• --- isomorphism, 302
• growth factor, 22
• Guldin's first rule, 453
• Guldin's second rule, 453
1
• H
2
• H
• half-angle formulas
• --- plane trigonometry, 142
• --- spherical trigonometry, 165
• half-line, notion, 128
• half-side formulas, 165
• Hamel basis, 599
• Hamilton
• --- circuit, 353
• --- differential equation, 815
• --- differential equation, partial, 801
3
• 841
2
• --- function, 520
3
• 801
2
• --- system, 815
• Hamiltonian, 538
• Hamming distance, 604
• Hankel transformation, 707
• harmonic analysis, 917
• harmonic, spherical
• --- first kind, 511
• --- second kind, 512
• Hasse diagram, 298
• heat conduction equation
• --- one-dimensional, 529
3
• 723
2
• --- three-dimensional, 537
• Heaviside
• --- expansion theorem, 719
• --- function, 641
3
• 697
2
• --- unit step function, 714
3
• 1084
2
• helix, 242
• Helmholtz, differential equation, 540
• Henon mapping, 798
3
• 812
2
• Hermite
• --- differential equation, 514
• --- polynomial, 514
3
• 545
2
• --- trapezoidal formula, 900
• Hessian
• --- matrix, 869
• --- normal form
• --- line equation, plane, 195
• --- plane equation, space, 214
• hexadecimal
• --- number, 937
• --- system, 937
• Hilbert
• --- boundary value problem, 593
• homogeneous, 593
• --- inhomogeneous, 594
• --- matrix, 989
• --- space, 615
• --- isomorphic, 618
• --- pre-Hilbert space, 615
• --- scalar product, 615
• histogram, statistics, 772
• hodograph, vector function, 642
• Holder
• --- condition, 591
• --- continuity, 591
• --- inequality, 32
• --- integrals, 32
• --- series, 32
• Holladay, theorem, 932
• holoedry, 311
• homeomorphism
• --- and topological equivalence, 810
• --- conjugate, 813
• --- orientation preserving, 843
• homomorphism, 302
3
• 341
2
• --- algebra, 614
• --- groups, 302
• --- linear operators, 600
• --- natural, 302
3
• 314
2
• --- ring, 314
• --- theorem, 341
• --- groups, 302
• --- ring, 314
• --- vector lattice, 603
• Hopf bifurcation, 831
• Hopf-Landau model, turbulence, 841
• Horner
• --- rule, 938
• --- scheme, 887
3
• 938
2
• --- two rows, 888
• horseshoe mapping, 827
• Householder
• --- method, 278
3
• 921
2
• --- tridiagonalization, 283
• --- l'Huilier equations, 167
• hull
• --- convex, 599
• --- linear, 597
• Hungarian method. 860
• hyperbola, 200
• --- arc, 203
• --- area, 202
• --- asymptote, 201
• --- binomial, 69
• --- conjugate, 202
• --- diameter, 202
• --- equation, 200
• --- equilateral, 64
3
• 203
2
• --- focal properties, 200
• --- focus, 200
• --- radius of curvature, 202
• --- segment, 202
• --- semifocal chord, 200
• --- tangent, 201
• --- transformation, 205
• --- vertex, 200
• hyperbolic
• --- cosecant, 87
• --- cosine, 87
• --- cotangent, 87
• --- secant, 87
• --- sine, 87
• --- tangent, 87
• hyperbolic function, 699
• --- geometric definition, 131
• --- inverse, complex, 699
• hyperboloid, 220
• --- one sheet, 220
3
• 222
2
• --- central surface, 224
• --- two sheets, 220
• --- central surface, 224
• hyperplane, 625
• --- of support, 626
• hypersubspace, 625
• hyp ersurface, 117
• hypocycloid, 102
• --- curtate, 102
• --- prolate, 102
• hypotenuse, 130
• hypothesis testing, 779
• hypotrochoid, 102
1
• I
2
• I
• icosahedron, --- infinite integration limits,
3
• icosahedro
• ideal, 313
• --- principal ideal, 313
• idempotence law
• --- Boolean algebra, 343
• --- propositional logic, 287
• --- sets, 293
• identically valid, 10
• identity, 10
• --- Boolean function, 344
• --- representation of groups, 303
• identity matrix, 253
• IEEE standard, 939
• IF-THEN rule, 372
• iff (if andonlyif), 320
• image
• --- function, 48
• --- set, 48
• --- space, 48
4
• 707
3
• --- subspace, 317
• imaginary
• --- number, 34
• --- part, 34
• --- unit, 34
• implication, 286
• --- proof, 5
• impulse function, 714
• impulse, rectangular, 714
• incidence function, 348
• incidence matrix, 350
• incircle
• --- quadrangle, 136
• --- triangle, 132
4
• 143
3
• incommensurability, 4
4
• 806
3
• increment, 21
4
• 394
3
• independence
• --- linear, 272
4
• 598
3
• --- path of integration, 468
4
• 687
3
• --- potential field, 662
• induction step, 5
• inequality, 28
• --- arithmetic and geometric mean, 30
• --- arithmetic and quadratic mean, 30
• --- Bernoulli, 30
• --- Bessel, 618
• --- binomial, 31
• --- Cauchy�Schwarz, 31
• --- Chebyshev, 31
4
• 754
3
• --- Chebyshev, generalized, 32
• --- different means, 30
• --- first degree, 33
• --- Holder, 32
• --- linear, 33
• --- first degree, 33
• --- solution, 33
• --- Minkowski, 32
• --- product of scalars, 31
• --- pure, 28
• --- quadratic, 33
• --- solution, 33
• --- Schwarz�Buniakowski, 615
• --- second degree, 33
• --- solution, 28
• --- special, 30
• --- triangle, 181
• --- norm, 611
• --- triangle inequality
• --- complex numbers, 30
• --- real numbers, 30
• infimum, 602
• infinite
• --- denumerable, 298
• --- non-denumerable, 298
• infinitesimal quantity, 441
• infinity, 1
• infix form, 298
• inflection point, 231
4
• 390
• 391
3
• initial conditions, 487
• --- hid
• initial value problem, 487
4
• 904
3
• inner product, 615
• inscribed circle
• --- quadrangle, 136
• --- triangle, 132
• inscribed pentagram. 138
• instability, round-off error
• --- numerical calculation, 942
• insurance mathematics, 21
• integer
• --- non-negative. 1
• --- part of x, 49
• --- programming, 847
• integrability
• --- complete, 521
• --- condition, 469
• --- conditions, 468
4
• 662
3
• --- differential, 468
• --- quadratic, 579
• integral
• --- absolutely convergent, 458
• --- antiderivative, 427
• --- basic
• --- notion, 428
• --- calculus, 427
• --- circuit, 462
4
• 468
3
• --- complex
• --- definite, 685
• indefinite, 686
• --- complex function
• --- measurable function, 637
• --- elementary functions, 428
• --- error integral, 461
• --- Euler, 459
4
• 461
3
• --- exponential integral, 461
• --- Fourier integral, 424
4
• 724
3
• --- Fresnel, 697
• interval of integration, 441
• --- Lebesgue integral, 637
• --- comparison with Riemann integral, 453
• --- limits of integration, 441
• --- line integral, 462
• --- first type, applications, 464
• --- lorarithmic integral, 431
• --- lower limit, 441
• --- non-elementary, 431
4
• 460
3
• --- parametric, 459
• --- primitive function, 427
• --- probability integral, 759
• --- Riemann, 441
• --- comparison with Stieltjes integral, 453
• --- Stieltjes, 628
• --- comparison with Riemann integral, 453
• --- notion, 453
• --- surface integral, 479
4
• 483
• 485
• 664
3
• --- triple integral, 476
• --- upper limit, 441
• --- volume integral, 476
• integral calculus, 427
• --- fundamental theorem, 442
4
• 458
3
• --- mean value theorem, 444
• integral cosine, 460
• integral curves, 798
• integral equation
• --- Abel, 590
• --- adjoint, 565
4
• 592
3
• --- approximation
• --- successive, 567
• --- characteristic, 592
• --- collocation method, 576
• --- eigenfunction, 565
• --- eigenvalue, 565
• --- first kind, 563
• --- Fredholm, 563
4
• 609
3
• --- degenerate kernel, 577
• --- first kind, 577
• --- second kind, 564
• --- general form, 563
• --- homogeneous, 563
• inhomogeneous, 563
• iteration method, 567
4
• 584
3
• --- kernel, 563
• --- degenerate, 564
• iterated, 587
• --- product, 564
• --- kernel approximation, 574
• --- tensor product, 574
• --- linear, 563
• --- Nystrom method, 573
• --- orthogonal system, 579
4
• 582
3
• --- perturbation function, 563
• --- quadratically integrable function, 579
• --- quadrature formula, 572
• --- second kind, 563
• --- singular, 590
• --- Cauchy kernel, 591
• --- transposed, 565
• --- Volterra, 563
4
• 609
3
• --- convolution type, 587
• --- first kind, 585
• --- second kind, 585
• integral equation method, closed curve, 534
• integral exponential function, table, 1051
• integral formula
• --- Cauchy, 689
• --- application, 694
• --- Gauss, 666
• --- series remainder, 418
• integral logarithm, table, 1053
• integral norm, 639
• integral surface, 518
• integral test of Cauchy, 408
• integral theorem, 665
• --- Cauchy, 688
• --- Gauss, 665
• --- Green, 666
• --- Stokes, 666
• integral transformation, 707
• --- application, 709
• --- Carson, 707
• --- definition, 707
• --- Fourier, 707
• --- Gabor. 743
• --- Hankel, 707
• image space, 707
• inverse, 707
• integral transformation (continued)
• --- kernel, 707
• --- Laplace, 707
• --- linearity, 707
• --- Mellin,707
• --- multiple, 709
• --- one variable, 707
• --- original space, 707
• --- several variables, 709
• --- special, 707
• --- Stieltjes, 707
• --- Walsh, 744
• --- wavelet, 740
4
• 741
3
• --- fast, 743
• integral, definite, 440
• --- differentiation, 443
• --- notion, 427
• --- particular integral, 442
• --- table, 1056
• --- algebraic functions, 1059
• --- exponential function, 1057
• --- logarithmic function, 1058
• --- trigonometric functions, 1056
• integral, elliptic, 437
• --- first kind, 431
4
• 437
• 703
3
• --- second kind, 437
• --- series expansion, 462
• --- table, 1061
• --- third kind, 437
• integral, improper, 440
• --- Cauchy's principal value, 454
4
• 457
3
• --- convergent, 454
4
• 456
3
• --- divergent, 454
4
• 456
3
• infinite integration limits, 453
2
• notion, --- vector,
3
• --- notion, 453
• --- principal value, 454
• --- unbounded integrand, 453
• integral, indefinite, 427
• --- basic integral, 428
• --- cosine function, table, 1043
• --- cotangent function, table, 1049
• --- elementary functions, 428
• --- elementary functions, table, 1023
• --- exponential function, table, 1051
• --- hyperbolic functions, table, 1050
• inverse hyperbolic functions, table, 1055
• inverse trigonometric function, table, 1054
• irrational functions, table, 1030
• --- logarithmic functions, table, 1053
• --- notion, 428
• --- other transcendental functions, table, 1050
• --- sine and cosine function, table, 1045
• --- sine function, table, 1041
• --- table, 1023
• --- tangent function, table, 1049
• --- trigonometric functions, table, 1041
• integral, logarithmic, 431
• integral, surface
• --- first type, 479
• --- general form, 485
• --- second type, 483
• integrand, 428
4
• 441
3
• integrating factor, 490
• integration
• --- approximate
• --- ordinary differential equation, 904
• --- partial differential equation, 911
• --- complex plane, 685
• --- constant, 428
• --- function, non-elementary, 460
• --- graphic, 432
• --- graphical, 446
• in complex, 694
• interval, 441
• --- limit
• --- depending on parameter, 459
• --- lower, 441
• --- upper, 441
• --- logarithmic, 430
• --- numerical, 898
• --- multiple integrals, 903
• --- partial, 431
• --- power, 430
• --- rational functions, 432
• --- rules
• --- by series expansion, 431
• --- by substitution, 431
• --- constant multiple rule, 429
• --- definite integrals, 443
• --- general rule, 429
• indefinite integrals, 430
• interchange rule, 444
• interval rule, 443
• --- series expansion, 446
4
• 460
3
• --- sum rule, 429
• --- under the integration sign, 459
• --- variable, 441
• --- variable, notion, 428
• --- vector field, 660
• --- volume, 450
• integrator, 447
• intensity, source, 668
• interaction, soliton, 547
• intercept theorem, 134
• interest, 22
• --- calculation, 21
• --- compound, 22
• intermediate value theorem
• --- one variable, 59
• --- several variables, 123
• intermediate variable, 382
• intermittence, 839
4
• 843
3
• Internationale Standard Book Number ISBN, 332
• interpolation
• --- Aitken-Neville, 918
• --- condition, 917
• --- formula
• --- Lagrange, 918
• --- Newton, 917
• --- fuzzy system, 377
4
• 378
3
• --- knowledge-based, 377
• --- node, 898
• --- equidistant, 904
• --- points, 917
• --- quadrature, 899
• --- spline, 917
4
• 931
3
• --- bicubic, 933
• --- cubic, 931
• --- trigonometric, 917
4
• 927
3
• interpretation
• --- formula, predicate logic, 289
• --- variable, 287
• intersection, 292
• --- angle, 159
• --- by two oriented lines, 147
• --- fuzzy set, 365
• --- on the sphere, 171
• --- point
• --- four planes, 216
• --- lines, 195
• --- plane and line, 218
• --- three planes, 216
• --- two lines, space, 219
• --- set, fuzzy set, 366
• --- sets, 292
• --- without visibility, 147
• interval
• --- convergence, 416
• --- numbers, 2
• --- order (O)-interval, 602
• --- rule, 443
• --- statistics, 772
• invariance
• --- rotation invariance, 265
• --- transformation invariance, 265
• --- translation invariance, 265
• invariant, 706
• --- quadratic curve, 206
• --- scalar, 212
• --- scalar invariant, 184
• --- surface second order, 224
• inverse, 601
• inverse function, 51
• --- hyperbolic, 91
• --- trigonometric, 84
• inverse transformation, 707
• inversion
• --- Cartesian coordinate system, 269
• --- conformal mapping, 676
• --- space, 269
• involute, 237
• irrational
• --- algebraic, 2
• --- quadratic, 2
• isometry, space, 611
• isomorphic
• --- vector spaces, 601
• isomorphism, 302
4
• 341
3
• --- Boolean algebra, 343
• --- graph, 349
• --- groups, 302
• --- surjective norm, 626
• iteration, 884
• inverse, 283
• --- method, 283
4
• 884
• 895
3
• --- ordinary, 884
4
• 896
3
• --- sequential steps, 896
• --- simultaneous steps, 896
• --- vector, 284
2
1
• J
2
• J
• Jacobi
• --- function, 703
• --- method, 283
3
• 895
2
• Jacobian
• --- determinant, 121
3
• 266
2
• --- matrix, 897
• Jordan
• --- matrix, 282
• --- normal form, 282
• jump, finite, 58
1
• K
2
• K
• KAM (Kolmogorov-Arnold-Moser) theorem, 815
• ker, 341
3
• 601
2
• kernel, 302
• --- approximation
• --- integral equation, 574
• --- spline approach, 575
• --- tensor product, 574
• --- congruence
• --- relation, 341
• --- homomorphism, 341
• --- integral equation, 563
• --- degenerate, 574
• --- iterated, 568
3
• 587
2
• --- resolvent, 570
• --- solving, 568
• --- integral transformation, 707
• --- operator, 601
• --- ring, 314
• --- subspace, 317
• key equation, 123
• kink soliton, 550
• Kirchhoff formula, 536
• Klein's four-group, 302
• Koch curve, 823
• Korteweg de Vries equation, 547
• Kronecker
• --- generalized delta, 265
• --- product, 258
• --- symbol. 253. 265
• Kuan, 352
• Kuhn-Tucker conditions, 862
• --- reference, 626
• Kuratowski theorem, 357
1
• L
2
• L
• l.c.m. (least common multiple), --- function,
3
• l.c.m. (
• Lagrange
• --- function, 403
• --- identity, vectors, 185
• --- interpolation formula, 918
• --- method of multiplier, 403
• --- theorem, 301
• Lagrangian, 403
4
• 862
3
• Laguerre
• --- differential equation, linear, second order, 513
• --- polynomial, 513
• Lanczos method, 283
• Landau, order symbol, 56
• Laplace
• --- differential equation, partial, 538
4
• 669
3
• --- expansion, 259
• --- wave equation, 536
• Laplace operator
• --- different coordinates, 657
• --- polar coordinates, 401
• --- vector components. 659
• Laplace transformation, 710
• --- addition law, 711
• --- comparing to Fourier transformation, 730
• --- convergence, 710
• --- convolution, 713
• --- one-sided, 713
• --- convolution, complex, 714
• --- definition, 710
• --- differential equation, ordinary
• linear, constant coefficients, 721
• linear, variable coefficients, 722
• --- differential equation, partial, 723
• --- differential equations, 721
• --- differentiation, image space, 712
• --- differentiation, original space, 712
• --- discrete, 736
• --- division law, 713
• --- frequency-shift theorem, 711
• --- image space, 710
• --- integration, image space, 712
• --- integration, original space, 712
• --- inverse, 710
4
• 718
3
• --- inverse integral, 720
• linearity law, 711
• --- original function, 710
• --- original space, 710
• --- partial fraction decomposition, 718
• --- piecewise differentiable function, 715
• --- series expansion, 719
• --- similarity law, 711
• --- step function, 714
• --- survey, 707
• --- table, 1067
• --- transform, 710
• --- translation law, 711
• largest area method, 374
• lateral area
• --- cone, 156
• --- cylinder, 154
• --- polyhedron, 151
• lateral face, 151
• latitude, geographical, 160
4
• 244
3
• lattice, 342
• --- Banach, 614
• --- Bravais, 310
• --- crystallography, 310
• --- distributive, 342
• --- vector, 614
• Laurent
• --- expansion, 692
• --- series, 692
4
• 736
3
• law
• --- cosine, 164
• --- spherical triangle, 164
• large numbers, 764
• --- Bernoulli, 764
• limit theorem, Lindeberg-Levy, 765
• --- sine-cosine, 164
• --- polar, 164
• layer, spherical, 157
• least common multiple (l.c.m.), integer numbers, 324
• least squares method, 107
4
• 277
• 890
• 909
• 913
• 919
3
• --- calculus of observations. 787
• --- Gauss, 403
• --- regression analysis, 780
• least squares problem, linear, 894
• Lebesgue
• --- integral, 454
4
• 637
3
• --- comparison with Riemann integral, 453
• --- measure, 636
4
• 816
3
• left singular vector, 285
• left-hand coordinate system, 207
• leg or side of an angle, 128
• Legendre
• --- differential equation, 511
• --- polynomial
• --- associated, 513
• --- first kind, 511
• --- second kind, 512
• --- symbol, 329
• Leibniz
• --- alternating series test, 410
• --- formula, 385
• lemma
• --- Jordan. 695
• --- Schur, 306
• lemniscate, 99
4
• 234
3
• length
• --- interval, 635
• line integral, first type, 464
• --- reduced, 174
• --- vector, 189
• length, arc
• --- space curve, 246
• level
• --- curve, 117
• line, 644
• --- surface, 644
• level set (fuzzy set), 364
• library (numerical methods), 944
• --- Aachen library, 946
• --- IMSL library,945
• --- NAG library, 944
• limit
• --- cycle, 806
4
• 833
3
• --- stable, 806
• --- unstable, 806
• --- definite integral, 440
• --- function
• --- complex variable, 671
• --- one variable, 51
• --- several variables, 122
• --- theorems, 53
• --- function series, 414
• --- integration, depending on parameter, 459
• --- partial sum, 414
• --- sequence of numbers, 405
• --- sequence, in metric space, 606
• --- series, 406
• --- superior, 416
• line, 62
• --- curvature, surface, 248
• --- geodesic, 159
• --- imaginary, 205
• --- notion, 128
• --- space, 150
4
• 214
3
• --- vector equation, 188
• line element
• --- surface, 245
• --- vector components, 660
• line equation
• --- plane, 194
• --- Hessian normal form. 195
• --- intercept form. 195
• --- polar coordinates, 195
• --- through a point, 194
• --- through two points, 194
• --- slope, plane, 194
• --- space, 217
• line integral, 462
• --- Cartesian coordinates, 661
• --- first type, 463
• --- applications, 464
• --- general type, 467
• --- second type, 464
• --- vector field, 660
• linear combination, vectors, 83
4
• 181
• 184
3
• linear form, 600
4
• 601
3
• --- continuous. 623
• linear programming
• --- assignment problem, 860
• --- basic notions, 849
• --- basic solution, 850
• --- basis of the extreme point, 849
• --- constraints, 846
• --- degenerate extreme point, 849
• --- distribution problem, 860
• --- dual problem, 856
• --- duality, 856
• --- extreme point, 849
• --- normal form, 849
4
• 850
3
• --- objective function, 846
• --- primal problem, 856
• --- round-tour problem, 861
• --- surplus variable, 850
• linearly
• --- dependent, 317
• --- independent, 317
• lines
• --- angle between, plane, 196
• --- intersection point, plane, 195
• --- orthogonal, 128
4
• 197
3
• --- parallel, 128
4
• 150
• 197
3
• --- pencil, 196
• --- perpendicular, 128
4
• 197
3
• --- skew, 150
• Liouville
• --- approximation theorem, 4
• --- formula, 501
4
• 802
3
• --- theorem, 801
• Lipschitz condition
• --- higher-order differential equation, 498
• --- ordinary differential equation, 488
• locus, geometric, 194
• logarithm
• --- binary, 10
• --- Briggsian, 9
• --- complex function, 678
• --- decimal, 9
• --- definition, 9
• --- natural, 9
4
• 698
3
• --- Neperian, 9
• --- principal value, 698
• --- table, 10
• --- taking of an expression, 9
• logarithmic decrement, 83
• logarithmic integral, 431
4
• 460
3
• logarithmic normal distribution, 759
• logic, 286
• --- fuzzy, 360
• --- predicate, 289
• --- propositional, 286
• longitude, geographical, 160
4
• 244
3
• loop function, 1007
• loop, graph, 348
• Lorentz curve, 731
• Lorenz system, 798
4
• 801
• 839
3
• loxodrome, 177
• --- arclength, 177
• --- course angle, 178
• --- intersection point, 178
• --- intersection point of two loxodromes, 179
• --- p space, 639
• LU factorization (lower and upper triangular matrix).
• --- 891
• Lyapunov
• --- exponent, 820
• --- function, 803
2
1
• M
2
• M
• MacDonald function, --- identity,
3
• MacDonald
• Maclaurin
• --- formula, 415
• --- series expansion, 418
• Macsyma (computer algebra system), 953
• majorant series, 413
• manifold
• --- center manifold theorem
• --- local, 835
• mappings, 835
• --- of solutions, 890
• --- stable, 806
4
• 812
3
• --- unstable, 806
4
• 812
3
• manipulation
• --- algebraic expressions, 979
• --- non-polynomial expressions, 981
• mantissa, 10
4
• 939
3
• manyness set, 298
• Maple (computer algebra system), 953
• Maple
• --- addendum to syntax, 978
• --- algebraic expressions, 971
• manipulation, 981
• multiplication, 981
• --- array, 973
• --- attribute, 978
• --- basic structure elements, 968
• --- context, 978
• --- conversion of numbers, different bases, 970
• Maple (continued)
• --- differential equations, 997
• --- differential operators, 977
• --- differentiation, 995
• --- eigenvalues, eigenvectors, 992
• --- elements of linear algebra, 989
• --- environment variable, 978
• --- equation
• --- one unknown, 986
• --- transcendental, 987
• --- factorization of polynomials, 982
• --- floating-point number, conversion, 970
• --- formula manipulation, introduction, 953
• --- functions, 975
• --- graphics, 1005
• --- introduction, 955
• --- three-dimensional, 1008
• --- two-dimensional, 1005
• --- help and information, 978
• --- input, output, 949
4
• 969
3
• --- input�output, 953
• --- integral
• --- definite, 997
• --- indefinite, 996
• multiple, 997
• --- lists, 972
• manipulation, general expressions, 983
• matrices, 973
• --- numerical calculations, introduction, 954
• --- numerical mathematics, 949
• --- differential equations, 952
• --- equations, 950
• --- expressions and functions, 949
• --- integration, 951
• --- object classes, 968
• --- objects, 968
• --- operations
• --- important, 986
• --- polynomials, 983
• --- operators
• --- functional, 976
• --- important, 971
• --- partial fraction decomposition, 983
• --- procedures, 975
• --- programming, 977
• --- sequences, 972
• --- short characteristc, 953
• --- special package plots, 1007
• --- system description, 968
• --- system of equations, 987
• --- systems of equations, linear, 990
• --- table structures, 973
• --- types, 968
• --- types of numbers, 970
• --- vectors, 973
• mapping, 294
4
• 296
3
• --- between groups, 302
• --- bijective, 296
4
• 600
3
• --- complex number plane, 684
• --- conformal, 674
• --- circular transformation, 676
• --- exponential function, 679
• --- fixed-point, 675
4
• 676
3
• --- inversion, 676
• --- linear fractional, 676
• --- linear function, 675
• --- logarithm, 678
• --- quadratic function, 677
• --- square root, 677
• --- sum of linear and fractional linear functions, 678
• --- contracting, 608
• --- equivalent, 843
• --- function, 48
4
• 294
3
• --- Henon mapping, 812
• --- horseshoe, 827
• --- injective, 296
4
• 600
3
• --- inverse mapping, 296
• --- kernel, 302
• --- lifted, 843
• --- linear, 317
4
• 319
• 600
3
• modulo, 816
• --- one-to-one, 296
• --- Poincare, 827
• --- Poincare mapping, 808
• --- reduced, 835
• --- regular, 318
• --- rotation mapping, 817
• --- shift, 820
• --- surjective, 296
4
• 600
3
• --- tent, 816
• --- topological conjugate, 813
• --- unit circle, 843
• marginal distribution, 754
• mass
• --- double integral, 474
• --- line integral, first type, 464
• --- triple integral, 480
• matching, 356
• maximal, 356
• --- perfect, 356
• --- saturated, 356
• Mathcad (computer algebra system), 953
• Mathematica (computer algebra system), 953
• Mathematica
• --- 3D graphics, 1004
• --- algebraic expressions, manipulation, 979
• --- algebraic expressions, multiplication, 979
• --- apply, 965
• --- attribute, 966
• --- basic structure element, 956
• --- context, 966
• --- curves
• --- parametric representation, 1003
• --- two-dimensional, 1002
• --- differential and integral calculus, 992
• --- differential equation, 994
• --- differential quotient, 992
• --- differentiation, 965
• --- eigenvalues, eigenvectors, 989
• --- elements, 956
• --- elements of linear algebra, 988
• --- equation, 985
• manipulation, 984
• --- transcendent, 985
• --- expression, 956
• --- factorization, polynomials, 980
• Mathematica (continued)
• --- FixedPoint, 965
• --- FixedPointList, 965
• --- floating-point number, conversion, 958
• --- formula manipulation, introduction, 953
• --- function, inverse, 965
• --- functional operations, 964
• --- functions, 963
• --- graphics, 998
• --- functions, 1001
• --- introduction. 955
• --- options, 1000
• --- primitives, 998
4
• 999
3
• --- head,956
• --- input and output, 946
• --- input-output, 953
4
• 957
3
• --- integral
• --- definite, 994
• --- indefinite, 993
• multiple, 994
• --- lists, 959
• manipulation with matrices. 962
• manipulation with vectors. 962
• Map, 965
• matrices as lists, 961
• messages, 967
• --- Nest, 965
• --- NestList, 965
• --- numerical calculations, introduction, 954
• --- numerical mathematics, 946
• --- curve fitting, 946
• --- differential equations, 948
• --- integration, 948
• --- interpolation, 947
• --- polynomial equations, 947
• --- objects, three-dimensional, 1004
• --- operations, polynomials, 980
• --- operators, important, 959
• --- partial fraction decomposition, 980
• --- pattern, 964
• --- programming, 966
• --- short characteristc, 953
• --- surfaces, 1004
• --- surfaces and space curves, 1003
• --- syntax, additional information, 966
• --- system description, 956
• --- system of equations, 986
• --- general case, 988
• --- special case, 988
• --- types of numbers, 957
• --- vectors as lists, 961
• mathematics, discrete, 286
• matrices
• --- arithmetical operations, 254
• --- associative law, 254
• --- commutative law, 254
4
• 255
3
• --- distributive law, 254
• --- division, 254
• --- eigenvalue, 278
• --- eigenvalue problem, 278
• --- eigenvectors, 278
• --- equality, 254
• multiplication, 254
• --- powers, 258
• --- rules of calculation, 257
• matrix, 251
• --- adjacency, 350
• --- adjoint, 251
4
• 260
3
• --- admittance, 355
• --- anti-Hermitian, 253
• --- antisymmetric, 252
• --- block tridiagonal, 912
• --- complex, 251
• --- conjugate, 251
• --- deflation, 284
• --- degree, 355
• --- diagonal, 252
• --- distance, 351
• --- extended coefficient, 892
• --- full rank, 277
• --- fundamental, 802
• --- Hermitian, 253
• --- Hessian, 869
• --- identity, 253
2
• incidence, --- triple integral,
3
• --- incidence, 350
• --- inverse, 256
4
• 260
3
• --- inverse calculation, 272
• --- Jacobian, 897
• --- Jordan,282
• --- lower triangular, 892
• main diagonal element, 252
• monodromy, 803
4
• 812
3
• --- normal, 252
• --- of coefficients, 272
• --- augmented, 273
• --- orthogonal, 257
• --- rank, 255
• --- real, 251
• --- rectangular, 251
• --- regular, 256
• --- rotation, 257
• --- scalar, 252
• --- self-adjoint, 253
• --- singular, 256
• --- size, 251
• --- skew-symmetric, 252
• --- spanning tree theorem, 355
• --- sparse, 912
• --- spur, 252
• --- square, 251
4
• 252
3
• --- stochastic, 766
• --- symmetric, 252
• --- trace, 252
• --- transposed, 251
• --- triangle decomposition, 890
• --- triangular, 253
• --- unitary, 257
• --- upper triangular, 892
• --- zero, 251
• matrix exponential function, 802
• matrix product, disappearing, 257
• max-min composition, 371
• maximum
• --- absolute, 60
4
• 123
• 390
3
• --- global, 390
• --- point, 847
• maximum (continued)
• --- relative, 390
• maximum-criterion method, 373
• Maxwell, diagonal method, 684
• mean
• --- arithmetic, 19
4
• 753
• 790
3
• --- estimate, 771
• --- geometric, 20
• --- golden, 845
• --- harmonic, 20
• --- quadratic, 20
• --- sample function, 771
• --- statistics, 773
• --- value, 19
• --- weighted, 753
4
• 790
• 793
3
• mean squares problem
• --- different versions, 403
• --- non-linear, 922
• mean squares value problem
• --- linear, 277
• --- rank deficiency case, 278
• mean value formula, 899
• mean value function, integral calculus, 444
• mean value method, 107
• mean value theorem
• --- differential calculus, 389
• --- generalized, 390
• --- integral calculus, 444
• --- generalized, 445
• measure
• --- concentrated on a set, 816
• --- counting, 635
• --- degrees, 130
• --- dimension, 824
• --- Dirac, 636
4
• 816
3
• --- ergodic, 817
• --- function, convergence theorems. 638
• --- Hausdorff, 822
• --- invariant, 816
• --- Lebesgue, 636
4
• 816
3
• --- natural, 816
• --- physical, 817
• --- probability, 638
• --- invariant, 816
• --- SBR (Sinai-Bowen-Ruelle), 817
• --- a-algebra, 635
• --- (T-finite, 638
• --- support, 816
• measured value, 787
• measurement
• --- error, 787
• --- error density, 788
• --- error, characterization, 788
• --- protocol, 788
• measuring error, distribution, 788
• median, 142
• --- sample function, 772
• --- statistics, 774
• --- triangle, 132
• Mellin transformation, 707
• Melnikov method, 841
• membership
• --- degree, 360
• --- function, 360
4
• 361
3
• --- bell-shaped, 362
• --- trapezoidal, 361
• --- relation, 290
• meridian, 160
4
• 244
3
• --- convergence, 169
• --- tangential, 176
• method
• --- Bairstow, 889
• --- barrier, 876
• --- Bernoulli, 889
• --- bisection, 283
• --- broken line, Euler, 904
• --- center of area, 374
• --- center of gravity, 373
• --- generalized, 374
• --- parametrized, 374
• --- Cholesky, 278
4
• 893
3
• --- collocation
• --- boundary value problem, 909
• --- integral equation, 576
• --- partial integral equation, 913
• --- cryptoanalysis, 337
• --- cryptography, 335
• --- cutting plane, 877
• --- descent, 868
• --- difference
• --- ordinary differential equations, 908
• --- partial differential equations, 911
• --- extrapolation, 905
• --- Fibonacci, 868
• --- finite difference, 534
• --- finite element, 534
4
• 913
3
• --- Galerkin, 909
• --- Gauss elimination, 890
• --- Gauss-Newton, 897
• --- derivative free, 898
• --- Gauss-Seidel, 895
• --- gradient. 561
• --- Graeffe, 889
• --- Green, 531
4
• 532
3
• --- Hildreth-d'Esopo, 867
• --- Householder, 278
4
• 921
3
• --- Hungarian, 860
• --- inner digits of squares, 783
• --- integrating factor, 490
• --- iteration, 283
4
• 884
• 890
• 895
3
• --- ordinary, 884
4
• 896
3
• --- Jacobi, 283
4
• 895
3
• --- Kelley's, optimization, 877
• --- Lagrange multiplier, 862
• --- Lanczos, 283
• --- largest area. 374
• --- least squares, 107
4
• 890
• 909
• 913
• 919
• 921
3
• Mamdani, 374
• maximum-criterion, 373
• mean-of-maximum, 373
• Monte Carlo, 783
• multi-step, 905
• multiple-target, 910
• --- Newton, 885
4
• 897
3
• modified, 885
• --- non-linear operators, 632
• method (continued)
• --- operator, 709
• --- orthogonalization, 278
4
• 893
• 894
3
• --- parametrized center of area, 374
• --- pivoting, 271
• --- polygonal, Euler, 904
• --- predictor-corrector, 906
• --- public key, 338
• --- regula falsi, 886
• --- relaxation, 896
• --- Riemann, 530
• --- Ritz, 560
4
• 909
3
• --- Romberg, 901
• --- RSA encryption, 339
• --- Runge-Kutta, 904
• --- separation of variables, 489
• --- shooting, 910
• --- single-target, 910
• --- SOR (successive overrelaxation), 896
• --- statistical experiment, 787
• --- steepest descent, 869
• --- Steffensen, 886
• --- successive approximation
• --- Banach space, 621
• --- Picard, 496
• --- Sugeno, 375
• --- transformation, 283
• --- undetermined coefficients, 16
4
• 497
3
• --- variation of constants, 501
• --- Vigenere cypher, 336
• --- Wolfe, 865
• metric, 604
• --- Euclidean, 604
• --- surface, 247
• Meusnier, theorem, 247
• midpoint line segment
• --- plane, 192
• --- space, 213
• midpoint rule, 906
• minimal surface, 249
• minimum
• --- absolute, 60. 123
4
• 390
3
• --- global, 390
• --- point, 862
• --- global, 862
• --- local, 862
• --- problem, 847
• --- relative, 390
• Minkowski
• --- inequality, 32
• --- integrals, 32
• --- series, 32
• mixed product, 187
• modal value, statistics, 774
• mode, statistics, 774
• model
• --- Hopf-Landau, turbulence, 841
• --- urn model, 755
• module of an element, 603
• modulo
• --- congruence, 297
4
• 327
3
• mapping, 816
• modulus, analytic function, 672
• --- de Moivre formula
• --- complex number, 37
• --- hyperbolic functions, 90
• --- trigonometric functions, 79
• Mollweide equations, 142
• moment of inertia, 263
4
• 452
3
• --- double integral, 474
• --- line integral, first type, 464
• --- triple integral, 480
2
• moment, order n, --- mutation,
3
• moment, order n, 753
• monodromy matrix, 803
4
• 804
• 812
3
• monotonicity, 635
• monotony, 388
• --- function, 49
• --- sequence, numbers, 404
• Monte Carlo method, 783
• --- application in numerical mathematics, 785
• --- usual, 785
• Monte Carlo simulation, 783
• --- example, 784
• Morse-Smale system, 815
• multi-index, 613
• multi-scale analysis, 743
• multi-step method, 905
• multiple integral, 471
• --- transformation, 709
• multiple-target method, 910
• multiplication
• --- complex numbers, 36
• --- computer calculation, 940
• --- polynomials, 11
• --- rational numbers, 1
• multiplicity, of divisors, 320
• multiplier,490,
4
• 803,
• 804,
• 812
3
• --- Lagrange method, 403
• mutation, 870
2
1
• N
2
• N
• nabla operator
• --- definition, 656
• --- divergence, 656
• --- gradient, 656
• --- rotation, 656
• --- twice applied, 657
• --- vector gradient. 656
3
• 657
2
• NAND function, 288
• nautical, radio bearing, 171
• navigation, 172
• negation, 286
• --- Boolean function, 344
• --- double, 288
• neighborhood, 605
• --- point, 605
• Neper
• --- equations, 167
• --- logarithm, 9
• --- rules, 169
• net
• --- chart, 123
• --- more than three variables, 127
• --- three variables, 123
• --- isometric, 675
• --- points, 933
• Neumann
• --- problem, 670
• --- series, 568
3
• 587
2
• Newton
• --- field, point-like mass, 646
• --- interpolation formula, 917
• Newton method, 885
3
• 897
2
• --- adaptive, 951
• --- modified, 885
• non-linear operators, 632
• --- approximation sequence, 632
• --- modified, 632
• non-linear optimization, 869
• --- damped, 869
• niveau line, 117
• nodal plane, 541
• node, 809
• --- approximation interval, 931
• --- differential equation, ordinary, 494
• --- saddle node, 804
3
• 809
2
• --- stable, 804
• --- triple, 834
• nominal rate, 22
• nomogram, 123
• nomography, 123
• non-negative integer, 1
• NOR function, 288
• norm
• --- axioms, linear algebra, 258
• --- Euclidean, 318
• --- integral, 639
• --- isomorphism, 626
• --- matrix, 258
• --- column sum norm, 259
• --- row sum norm, 259
• --- spectral norm, 259
• --- subordinate, 259
• --- uniform norm, 259
• --- operator norm, 619
3
• 802
2
• --- pseudonorm, 624
• --- residual vector, 277
• --- seminorm, 624
• --- .s'-norm, 365
• --- space, 611
• --- t-norm, 365
• --- vector, 258
• --- Euclidean norm, 258
• --- matrix norm, 258
• --- sum norm, 258
• normal distribution, 758
• --- logarithmic normal distribution, 759
• --- observational error, 72
• --- standard normal distribution, 759
• --- standard, table, 1091
• --- two-dimensional, 780
• normal equation, 894
3
• 919
• 921
2
• --- system, 782
3
• 920
2
• normal form, 345
• --- equation of a surface, 220
• --- Jordan,282
• --- principal conjunctive, 345
• --- principal disjunctive, 345
• normal plane, space curve, 238
3
• 241
2
• normal vector
• --- plane, 214
• --- plane sheet, 663
• --- surface, 244
• normal, plane curve, 226
• normalization condition, 541
• normalizing factor, 195
• northern direction
• --- geodesical, 169
• --- geographical, 169
• northwest corner rule, 858
• notation
• --- Polish notation, 355
• --- postfix notation, 355
• --- prefix notation, 355
• --- reversed polish, 355
• --- /7,-tuple, 117
• number, 1
• --- approximation, 4
• --- cardinal, 291
3
• 298
2
• --- complex, 34
• --- absolute value, 35
• --- addition, 36
• --- algebraic form, 34
• --- argument, 35
• --- division, 37
• --- exponential form, 35
• --- main value, 35
• --- multiplication, 36
• --- power, 37
• --- subtraction, 36
• --- taking the root, 38
• --- trigonometric form, 35
• --- composite, 320
• --- conjugate complex, 36
• --- imaginary, 34
• --- integer, 1
• --- interval, 2
• --- irrational, 2
3
• 297
2
• natural, 1
• non-negative, 1
• --- random, 770
• --- rational, 1
3
• 3
2
• --- real, 2
• --- taking of root, 8
• --- sequence, 404
• --- convergent, 606
• --- metric space, 606
• --- transcendental, 2
• number line, extended, 635
• number plane, complex, Gauss, 34
• number representation, computer internally, 936
3
• 938
2
• number system, 936
• --- binary, 937
• --- decimal, 937
• --- hexadecimal, 937
• --- octal, 937
• number theory, elementary, 320
• numbers
• --- Bernoulli. 412
• --- Euler, 412
• --- Fermat, prime, 321
• --- Fibonacci, 325
3
• 845
2
• --- Mersenne, prime, 321
• --- prime. 320
• numerical
• --- analysis, 884
• --- axis, 1
• --- calculations, computer, 939
• --- library (numerical methods), 884
• nutation angle, 211
• Nystrom method. 573
1
• O
2
• O
• obelisk, 153
• objective function, linear, 846
• observational value, 787
• occupancy number, statistics, 772
• octahedron, 154
• octal number, 937
• octal system, 937
• octantj 209
• --- Lj-limit set, 799
3
• 805
• 812
2
• operation, 298
• --- algebraic, 263
• --- arithmetical, 1
• --- associative, 298
• --- binary, 298
• --- commutative, 298
• --- exterior, 299
• --- n-ary, 298
• operational method, 534
• operational notation, differential equation, 502
• operator
• --- adjoint, 626
• --- AND, 368
• --- bounded, 619
• --- closed, 620
• --- coercivity, 634
• --- compact, 628
• --- compensatory, 368
• --- completely continuous, 628
• --- continuous, 610
• --- inverse, 621
• --- contracting, 608
• --- demi-continuous, 634
• --- differentiable, 632
• --- finite dimensional, 628
• --- gamma, 368
• --- Hammerstein, 631
• --- idempotent, 628
• --- inverse, 601
• --- isotone, 633
• --- lambda, 368
• --- linear, 319
3
• 600
2
• --- bounded, 619
• --- continuous, 619
• --- linear, notion, 319
• --- linear, permutability, 319
• --- linear, product, 319
• --- monotone, 634
• --- Nemytskij, 631
• --- non-linear, 631
• --- norm, 802
• --- notion, 48
• OR, 368
• --- positive, 602
• --- positive definite, 628
• --- positive non-linear, 633
• --- self-adjoint, 627
• --- singular, 592
• --- strongly monotone, 634
• --- Urysohn, 632
• operator method, 709
• opposite side, 130
• optimality condition, 862
3
• 863
2
• --- sufficient, 862
• optimality principle, Bellman, 880
• optimization, 846
• optimization method
• --- Bellman functional equation, 881
• --- conjugate gradient, 869
• --- cutting plane method, 877
• --- damped, 869
• --- DFP (Davidon-Fletcher-Powell), 870
• --- feasible direction, 872
• --- Fibonacci method, 868
• --- golden section, 868
• --- Hildreth-d'Esopo, 867
• --- Kelley, 877
• --- penalty method, 875
• --- projected gradient, 873
• --- unconstrained problem, 868
• --- Wolfe, 865
• optimization problem
• --- convex, 625
3
• 863
2
• --- dynamic, 878
• --- non-linear, 926
• optimization, evolution strategies, 870
• --- mutation, 870
• --- recombination, 870
3
• 871
2
• --- selection, 870
• optimization, non-linear, 861
• --- barrier method, 876
• --- convex, 863
• --- convexity, 864
• --- descent method, 868
• --- direction search program, 872
• --- duality, 863
• --- gradient method
• --- inequality constraints, 871
• --- Newton method, 869
• --- numerical search procedure, 867
• --- quadratic, 864
• --- saddle point, 862
• --- steepest descent method, 869
• orbit, 797
• --- double, periodic, 836
• --- heteroclinic, 808
• --- homoclinic, 808
3
• 841
2
• --- periodic, 797
• --- hyperbolic, 805
• --- saddle type, 805
• order
• --- (O)-interval, 602
• --- curve, second order, 205
• --- curve, n-th order, 194
• --- differential equation, 487
• --- interval, 602
• of magnitude, function, 55
• order (continued)
• --- second order surface, 220
• --- wavelet, 741
• order relation, 296
• --- linear, 297
• order symbol, Landau. 56
• ordering, 297
• --- lexicographical, 297
• --- linear, 297
• --- partial 297
3
• 601
2
• ordinate
• --- plane coordinates, 189
• --- space coordinates, 208
• Ore theorem, 353
• orientation, 269
• --- coordinate system, 207
• --- numerical axis, 1
• origin
• --- numerical axis, 1
• --- plane coordinates, 189
• --- space coordinates, 208
• original function, 707
• original space, 707
• --- set, 48
• orthocenter, triangle, 132
• orthodrome, 173
• --- arclength, 174
• --- course angle, 173
• --- intersection point, 174
3
• 175
2
• --- intersection point of two orthodromes, 178
• --- point, closest to the north pole, 173
• orthogonal
• --- function system, 920
• --- polynomials, 920
• --- space, 616
• --- spherical, 169
• orthogonality
• --- conditions
• --- line-plane, 219
• --- lines in space, 219
• --- planes, 216
• --- eigenvalues, differential equation, 515
• --- Hilbert space, 616
• --- lines, 128
• --- real vector space, 318
• --- trigonometric functions, 318
• --- vectors, 184
• --- weight, 515
• orthogonalization method, 893
3
• 894
2
• --- Givens, 895
• --- Gram-Schmidt, 617
• --- Householder, 278
3
• 895
2
• --- Schmidt, 579
• orthogonalization process, 280
• --- Gram-Schmidt, 280
• orthonormal
• --- function system, 920
• orthonormalization, vectors, 262
• oscillation
• --- duration, 82
• --- harmonic, 82
• oscillator, linear harmonic, 544
• osculating plane, space curve, 238
3
• 241
2
• osculation point, curve, 232
• oversliding, 266
1
• P
2
• P
• pair, ordered, --- irreducible,
3
• pair, ordered
• parabola, 203
• --- arclength, 204
• --- area, 204
• --- axis, 203
• --- binomial, 69
• --- cubic, 63
• --- diameter, 204
• --- directrix, 203
• --- equation, 203
• --- focus, 203
• --- intersection figure, 221
• --- n-th degree, 64
• parameter, 203
• --- quadratic polynomial, 62
• --- radius of curvature, 204
• --- semicubic, 93
• --- semifocal chord, 203
• --- tangent, 204
• --- transformation, 205
• --- vertex, 203
• paraboloid, 221
• --- elliptic, 221
• --- hyperbolic, 222
• --- central surface, 224
• --- invariant signs
• --- elliptic, 224
• --- hyperbolic, 224
• parabolic, 224
• --- of revolution, 222
• parallel circle, 244
• parallelepiped, 152
• --- rectangular, 152
• parallelism
• --- lines, 128
• parallelism conditions
• --- line-plane, 219
• --- lines in space, 219
• planes, 216
• parallelogram, 134
• parallelogram identity, unitary space, 615
• parameter, 11
4
• 49
3
• parabola, 203
• --- statistical, 773
• parameter space, stochastic, 765
• parametric integral, 459
• parametric representation, circle, 197
• parametrized center of area method, 374
• parity, 544
• Parseval
• --- equation, 422
4
• 516
• 579
• 618
3
• --- formula, 729
• partial fraction decomposition, 15
• --- special cases, 1029
• partial ordering, 297
• partial sum, 406
• partition, 297
• Pascal limacon, 96
• Pascal triangle, 13
• path, 351
• --- closed, 351
• --- integration, 463
• PCNF (principal conjunctive normal form), 345
• PDNF (principal disjunctive normal form), 345
• pencil of lines, 196
• pendulum
• --- equation, 841
• --- Foucault, 949
• --- mathematical, 702
• period, 703
• pentagon, regular, 138
• pentagram, 4
• pentagram, regular, 138
• percentage, 21
• --- calculation, 21
• performance score, 372
• perimeter
• --- circle, 138
• --- ellipse, 200
• polygon, 137
• period, 50
• --- secant, 76
• --- sine, 75
4
• 82
3
• --- tangent, 76
• period doubling, 836
• --- cascade, 837
4
• 842
3
• period parallelogram, 704
• permutability, linear operators, 319
• permutation, 745
• --- cyclic, vectors, 207
• --- group, 301
• --- matrix, 892
• --- with repetition, 745
• --- without repetition, 745
• perturbation, 535
4
• 941
3
• Pesin formula, 821
4
• 828
3
• phase
• --- initial, 82
• portrait, dynamical systems. 800
• --- shift, 75
4
• 82
3
• --- sine, 75
4
• 82
3
• --- space, dynamical systems, 797
• --- spectrum, 727
• physical units, 1012
• Picard
• --- iteration method
• --- integral equation, 567
• --- successive approximation method, 496
• Picard-Lindelof theorem, 610
• pivot, 891
• --- column, 271
4
• 852
3
• --- element. 271. 852
• --- row, 271
4
• 852
3
• --- scheme, 271
• pivoting, 271
4
• 274
3
• --- column pivoting, 892
• --- step, 271
• plane
• --- equation
• --- general, 214
• --- in space, 214
• --- geometry, 128
• --- intersecting, 158
• --- rectifying, 239
4
• 241
3
• --- space, 150
4
• 214
3
• --- vector equation, 188
• planes
• --- orthogonality, conditions, 216
• parallel, 150
• --- distance, 217
• parallelism, conditions, 216
• planimeter, 447
• Poincare mapping, 808
4
• 810
3
• Poincare section, 841
• point
• --- accumulation, 606
• --- asymptotic, curve, 232
• --- boundary, 606
• --- circular, 249
• --- coordinates, 189
• --- corner, 232
• --- cuspidal, curve, 232
• --- discontinuity, 57
• --- double, curve, 232
• --- fixed, conformal mapping, 676
• --- fixed, stable, 836
• --- focal, ordinary differential equation, 495
• --- improper, 192
• --- infinite, 192
• --- interior, 605
• --- isolated, 606
• --- isolated, curve, 232
• --- limit, 237
• --- limiting, 606
• --- multiple, curve, 233
• --- /(.-dimensional space, 117
• --- nearest approach, 237
• --- neighborhood, 605
• --- non-wandering, 815
• --- notion, 128
• plane curve, 231
• --- rational, 1
• --- regular, surface, 244
• --- saddle, ordinary differential equation, 495
• --- singular, 225
4
• 232
3
• --- isolated, 494
• --- ordinary differential equation, 494
• --- surface, 244
4
• 245
3
• --- spectrum, 622
• --- spiral, ordinary differential equation, 495
• --- surface point
• --- circular, 249
• --- elliptic, 248
• --- hyperbolic, 249
• parabolic, 249
• --- spherical, 248
• --- umbilical, 249
• --- terminal, curve, 232
• --- transversal homoclinic, 812
• --- umbilical, 249
• Poisson
• --- differential equation, partial, 538
4
• 667
• 670
3
• --- distribution, 757
• --- formula, 536
• --- integral, 533
• polar, 162
• --- angle, 190
• --- axis, 190
• --- coordinates, plane, 190
• --- coordinates, spherical, 209
• --- coordinates, transformation, 191
• --- distance, 244
• --- equation, 542
• --- curve second order, 207
• --- subnormal, 227
• --- subtangent, 227
• pole
• --- complex function, 673
• --- function, 58
• --- left, 162
• --- multiplicity m, complex function, 693
• --- on the sphere, 162
• --- order m, complex function, 693
• --- origin, 190
• --- right, 162
• Polish notation, 355
• polyeder, 151
• polygon
• --- area
• --- 2n-gon, 138
• --- n-gon, 137
• --- base angle, 137
• --- central angle, 137
• --- circumcircle radius, 137
• --- circumscribing, 137
• --- exterior angle, 137
• --- inscribed circle radius, 137
• --- interior angle, 137
• perimeter, 137
• plane, 137
• --- regular, convex, 137
4
• 138
3
• --- side length, 137
• --- similar, 133
• polyhedral angle, 151
• polyhedron, 151
• --- convex, 154
• --- regular, 154
• polynomial, 62
• --- Bernstein, 935
• --- characteristic, 278
• --- Chebyshev formula, 87
• --- division algorithm, 315
• --- division with remainder, 315
• --- equation, numerical solution, 887
• --- first degree, 62
• --- Hermite, 545
4
• 617
3
• --- integral rational function, 60
• --- interpolation, 917
• --- irreducible, 315
2
• Laguerre, --- right-angled triangle,
3
• --- Laguerre, 513
4
• 617
3
• --- Legendre, 511
4
• 616
3
• --- minimal, 316
• --- n-th degree, 63
• primitive, 315
• product representation, 43
• --- quadratic, 62
• --- ring, 315
• --- second degree, 62
• --- third degree, 63
• --- trigonometric, 927
• polynomials. 11
• Chebyshev, 924
• --- orthogonal, 920
• population, 769
• --- two-stage, 754
• Posa theorem, 353
• position coordinate, reflection, 269
• positive definite, 893
• postfix notation, 355
• postman problem, Chinese, 352
• potential
• --- complex, 681
• --- equation, 538
• --- field, 662
• --- conservative, 662
• --- rotation, 656
• --- retarded, 537
• power
• --- complex number, 37
• --- notion, 7
• --- real number, 7
• --- reciprocal, 68
• power series, 416
• --- asymptotic, 419
4
• 420
3
• --- complex terms, 690
• --- expansion, analytic function, 689
• --- inverse, 417
• power set, 298
• power spectrum, 818
• precession angle, 211
• predicate, 289
• --- logic, 289
• --- n-ary, 289
• predictor, 906
• predictor-corrector method, 906
• prefixes, metric, 1012
• pre-Hilbert space, 615
• present value, 25
• pressure, 451
• prime
• --- coprime, 5
4
• 14
3
• --- decomposition, canonical, 321
• --- element
• --- numbers, 320
• polynomials, 315
• --- factorization, 321
• --- canonical, 321
• --- Fermat, 321
• --- Mersenne, 321
• --- notation (measurement protocol), 788
• --- number, 320
• pair, 321
• --- quadruplet, 321
• --- relatively, 14
• --- triplet, 321
• prime formula, predicate logic, 289
• principal (a sum of money), 21
• principal axis
• --- direction, 265
• --- transformation, 264
• principal ideal, 313
• principal normal
• --- section, surface, 248
• --- space curve, 239
4
• 241
3
• principal quantities, 11
• principal value
• --- integral, improper, 454
• --- inverse hyperbolic function, 700
• --- inverse trigonometric function, 84
4
• 700
3
• --- logarithm, 698
4
• 700
3
• principle
• --- Cauchy, 608
• --- contracting mapping, 608
• --- extensionality, 291
• --- extrapolation principle, 902
• --- Neumann, 307
• --- of bivalence, 286
• prism, 151
• --- lines, 151
• --- regular, 151
• probability
• --- area interpretation, 752
• --- conditional, 750
• --- definition, 749
• --- total, 750
• probability integral, 759
• probability measure, 816
• --- ergodic, 820
• --- invariant, 816
• probability paper, 774
• probability theory, 745
4
• 747
3
• probability vector, 767
• problem
• --- Cauchy, 518
• --- Dirichlet, 528
4
• 670
3
• --- discrete, 921
• --- inhomogeneous, 537
• --- multidimensional
• --- computation of adjustments, 922
• --- Neumann, 670
• --- regularized, 278
• --- scheduling, 861
• --- shortest way, 351
• --- Sturm-Liouville, 515
• --- two-body, 520
• process
• --- birth process, 769
• --- death process, 769
• --- orthogonalization process, 280
• Poisson process, 768
• --- stochastic, 765
• product, 7
• --- algebraic, 366
• --- Cartesian
• --- fuzzy sets, 369
• --- n-ary, 370
• --- sets,294
• --- cross, 183
• --- derivative, 381
• --- direct
• --- group, 301
• --- ft-algebra, 341
• --- dot, 183
• --- drastic, 366
• --- dyadic, 255
• --- dyadic, tensors, 264
• --- inner product, 615
• --- Kronecker, 258
• --- n times direct, 343
• product sign, 7
• --- rules of calculation, 7
• --- scalar, 183
4
• 255
3
• --- vector, 183
• programming, 846
• --- computer algebra system, 955
• --- continuous dynamic, 878
• --- discrete dynamic, 878
• --- Bellman functional equation, 879
• --- Bellman functional equation method, 881
• --- Bellman optimality principle, 880
• --- constraint dynamic, 878
• --- constraint static, 878
• --- knapsack problem, 879
• problem, 878
• purchasing problem, 879
• --- state vector, 878
• --- linear, 846
• --- linear, transportation problem, 857
• --- Maple, 977
• --- Mathematica, 966
• problem
• --- dual problem, 856
• primal problem, 856
• programming, discrete
• --- continuous dynamical, 878
• --- cost function, 879
• --- decision, 878
• --- decision space, 878
• --- dynamic, 878
• --- functional equation, 880
• --- Bellman, 879
• --- method, 881
• --- knapsack problem, 879
4
• 882
3
• --- minimum interchangeability, 880
• --- minimum separability, 879
• --- n-stage decision process, 878
• --- optimal policy, 881
• --- optimal purchasing policy, 881
• purchasing problem, 879
• --- state costs, 878
• --- state vector, 878
• programming, linear
• --- constraints, 846
• --- forms, 846
• --- general form, 846
• properties, 848
• --- scheduling problem, 861
• programming, non-linear
• --- see optimization, non-linear, 861
• projection sides, 142
• projection theorem, orthogonal space, 616
• projector, 628
• proof
• --- by contradiction, 5
• --- constructive, 6
• --- direct, 5
• proof (continued)
• --- indirect, 288
• --- implication, 5
• --- mathematical induction, 5
• --- step from n to n + 1
4
• 5
3
• proportionality
• --- direct, 62
• --- inverse, 64
• proportions, 17
• proposition, 286
• --- dual, 343
• propositional logic, 286
• --- basic theorems, 287
• --- expression, 286
• propositional operation, 286
• --- extensional, 286
• propositional variable, 286
• protocoll, 772
• pseudonorm, 624
• pseudorandom numbers, 784
• pseudoscalar, 270
• pseudotensor, 268
4
• 270
3
• pseudovector, 268
4
• 270
3
• Ptolemy's theorem, 136
• pulse, rectangular, 697
• pyramid, 152
• --- frustum, 152
• --- n-faced, 152
• --- regular, 152
• --- right, 152
• --- truncated, 152
• Pythagoras
• --- general triangle, 142
• --- right-angled triangle, 141
2
1
• Q
2
• Q
• QR algorithm, 283
3
• 894
2
• QR decomposition, 894
• quadrangle, 134
3
• 136
2
• --- circumscribing, 136
• --- concave, 136
• --- convex, 136
• --- inscribed, 136
• quadrant relations, trigonometric functions, 77
• quadrant, Cartesian coordinates, 189
• quadratic
• --- curve, 205
• --- surface, 223
• quadratic form
• --- first fundamental, of a surface, 245
• --- index of inertia, 282
• --- real positive definite, 282
• --- second fundamental, surface, 248
• --- transformation
• --- principal axis, 281
• quadrature formula, 898
• --- Gauss type, 900
• --- Hermite, 899
• --- integral equation, 572
• --- interpolation quadrature, 899
• --- Lobatto type, 901
• quadruple (ordered four-tuple), 294
• quantification, restricted, 290
• quantifier, 289
• quantile, 752
• quantity, infinitesimal, 441
3
• 447
2
• quantum number, 541
• --- energy, 541
• --- magnetic, 544
• --- orbital angular momentum, 543
• --- vibration quantum number, 546
• quartic, 96
• quasiperiodic, 806
• queuing, 769
• --- theory, 769
• quintuple (ordered 5
3
• 294
2
• quotient, 1
3
• 12
2
• --- derivative, 382
• --- differential, 379
• --- set, 297
1
• R
2
• R
• radial equation, --- matrix calculation,
3
• radial
• radian
• --- definition, 130
• --- measure, 130
• radicals, ordinary differential equation, 493
• radicand, 8
• radius
• --- circle, 139
4
• 197
3
• --- circumcircle, 142
• --- convergence, 416
• --- curvature, 230
• --- curve, 229
• --- space curve, 240
• --- curvature, extremal, 558
• --- polar coordinates, 190
• --- principal curvature
• --- surface, 247
• --- short, 132
• --- torsion, space curve, 242
• --- vector, 181
• raising to a power
• --- complex numbers, 37
• real numbers, 7
4
• 9
3
• random number, 770
4
• 783
3
• --- application, 784
• --- congruence method, 783
• --- construction, 784
• --- different distributions, 784
• --- pseudorandom, 784
• --- uniformly distributed, 783
• random numbers, table, 1097
• random variable, 751
• --- continuous, 751
4
• 752
3
• --- discrete, 751
• --- independent, 754
4
• 779
3
• --- mixed, 751
• --- multidimensional, 754
• --- two-dimensional, 779
• random vector
• --- mathematical statistics, 770
• --- multidimensional random variable, 754
• random walk process, 786
• range, 47
• --- operator, 600
• --- sample function, 772
• --- statistics, 774
• rank
• --- matrix, 255
• --- tensor, 262
• --- vector space, 318
• rate
• --- effective, 22
• --- nominal, 22
• --- of interest, 22
• ray point, 495
• ray, notion, 128
• Rayleigh-Ritz algorithm, 283
• reaction, chemical, concentration, 116
• real part (complex number), 34
• rebate, 21
• recombination, 870
4
• 871
3
• rectangle, 135
• rectangular formula, simple, 899
• rectangular impulse, 714
• rectangular pulse, 697
• rectangular sum, 899
• rectification, 107
• Reduce (computer algebra system), 953
• reduction, 21
• --- angle, 174
• reduction formula, trigonometric functions, 77
• reflection principle, Schwarz, 681
• reflection, position coordinate, 269
• region, 118
• --- multiply-connected, 118
• --- non-connected, 118
• --- simply-connected, 118
• --- two-dimensional, 118
• regression
• --- analysis, 779
• --- coefficient, 781
• --- line, 780
4
• 781
3
• --- linear, 780
• --- multidimensional, 781
• regula falsi, 886
• regularity condition, 863
4
• 876
3
• regularization method, 285
• regularization parameter, 278
• relation, 294
• --- binary, 294
• --- congruence relation, 340
• --- equivalence relation, 296
• --- Euler (complex numbers), 35
4
• 698
3
• --- fuzzy-valued, 369
• --- inverse, 295
• --- less or equal than (< relation), 289
• --- matrix, 294
• --- n-ary, 294
• --- n-place, 294
• --- order relation, 296
• --- product, 295
• relaxation method, 896
• relaxation parameter, 896
• reliability testing, 769
• relief, analytic function, 672
• remainder
• --- estimation, 413
• --- series, 406
• --- term, 406
• Remes algorithm, 925
• rent, 25
• --- ordinary, constant, 25
• representation of groups, 303
• --- adjoint, 304
• --- direct product, 305
• --- direct sum, 305
• --- equivalent, 304
• --- faithful, 303
• --- identity, 303
• --- irreducible, 305
• --- non-equivalent, 304
• --- particular, 303
• --- properties, 303
• reducible, 305
• reducible, complete, 306
• representation matrix, 303
• representation space, 303
• --- subspace, 305
• --- true, 303
• --- unitary, 304
• representation theorem (fuzzy logic), 364
• resection
• --- Cassini, 149
• --- Snellius, 148
• residual spectrum, 623
• residual sum of squares, 277
4
• 894
3
• residual vector, 277
• residue
• --- quadratic modulo m, 329
• --- complex function, 693
• --- theorem, 694
• residue class, 327
• --- addition, 327
• --- multiplication, 327
• --- primitive, 328
• relatively prime, 328
• ring, 313
4
• 327
3
• ring, modulo m, 327
• residue theorem, 692
• --- application, 695
• residuum, 277
4
• 893
3
• resolvent, 568
4
• 570
• 622
3
• --- set, 622
• resonance torus, 842
• reversing the order of the bits, 930
• rhombus, 135
• Riemann
• --- formula, 530
• --- function, 530
• --- integral, 441
• --- comparison with Lebesgue integral, 453
• --- comparison with Stieltjes integral, 453
• --- method, 530
• --- sum, 441
• --- surface, many-sheeted, 685
• --- theorem, 409
• right screw rule, 663
• right singular vector, 285
• right-hand coordinate system, 207
• right-hand rule, 183
4
• 663
3
• ring, 313
• --- factor ring, 314
• ring (continued)
• --- homomorphism, 314
• --- theorem, 314
• --- isomorphism, 314
• --- polynomial, 315
• --- subring, 313
• risk theory, 21
• Ritz method, 560
4
• 909
3
• R', n-dimensional Euclidean vector space, 254
• Romberg method, 901
• root
• --- complex function, 673
• --- complex number, 38
• --- equation, 43
• --- non-linear, 884
• --- notion, 8
• --- JV-th of unity, 929
• real number, 8
• --- square root, complex, 677
• --- theorem of Vieta, 43
• root locus theory, 889
• root test of Cauchy, 408
• rotation
• --- definition, 654
• --- different coordinates, 655
• --- mapping, 817
• --- potential field, 656
• remark, 650
• --- vector components. 659
• --- vector field, 654
• rotation angle, 212
• rotation field
• --- pure, 668
• --- zero-divergence field, 668
• rotation invariance, 265
• rotation matrix, 190
4
• 211
• 257
3
• --- orthogonal, 262
• rotation number, 843
• rotator, rigid, 542
• round-off, 940
• --- error, 942
• --- measurement, 787
• --- error method, 942
• round-tour problem, 861
• row echelon form, system of linear equations, 276
• row sum criterion, 895
• RSA encryption method, 339
• Ruelle-Takens-Newhouse scenario, 842
• rule, 906
• --- Adams and Bashforth, 906
• --- Bernoulli-l'Hospital, 54
• --- Cartesian rule of signs, 888
• --- Cramer, 275
• --- De Morgan, 287
• --- Descartes, 45
• --- Guldin, first rule, 453
• --- Guldin, second rule, 453
• --- Laplace expansion, 259
• --- linguistic, 375
• --- Milne, 906
• --- Sarrus, 261
• ruled surface, 250
• rules
• --- composition, 371
• --- determinant calculation, 260
• --- divisibility, elementary, 320
• --- matrix calculation, 257
2
• Neper, --- Runge-Kutta method,
3
• --- Neper, 169
• --- tensor calculation, 263
• Runge-Kutta method, 904
2
1
• S
2
• S
• saddle, --- shift register, lineare,
3
• saddle, 804
4
• 811
3
• saddle form, 249
• saddle point
• --- differential equation, 495
• --- Lagrange function, 862
• sample, 754
4
• 770
3
• --- function, 770
• --- random, 770
• size, 754
• summarizing the sample, 772
• --- variable, 770
• Sarrus rule, 261
• scalar
• --- invariant, 263
• --- notion, 180
• scalar field, 643
• --- axial, 644
• --- central field, 643
• --- coordinate definition, 644
• --- directional derivative, 649
• --- gradient, 650
4
• 651
3
• --- plane, 643
• scalar matrix, 252
• scalar product, 183
4
• 318
3
• --- Hilbert space, 615
• --- representation in coordinates, 186
4
• 188
3
• --- rotation invariance property, 270
• --- two functions, 920
• --- vectors, 255
• scale
• --- cartography, 144
• --- equation, 114
• --- factor, 114
4
• 146
3
• --- logarithmic, 114
• --- notion, 114
• semilogarithmic, 115
• scenario, Ruelle-Takens-Newhouse, 842
• Schauder fixed-point theorem, 633
• scheduling problem, 861
• scheme
• --- Falk, 254
• --- Young, 306
• Schmidt, orthogonalization method, 579
• Schoenflies symbolism, 307
• Schrodinger equation
• --- linear, 538
• --- non-linear, partial, 547
4
• 549
3
• --- time-dependent, 539
• --- time-independent, 539
• Schur's lemma, 306
• Schwarz
• --- exchange theorem, 395
• --- reflection principle, 681
• Schwarz�Buniakowski inequality, 615
• Schwarz-Christoffel formula, 679
• screw
• --- left, 242
• --- right, 242
• search procedure, numerical, 867
• secant
• --- geometric definition, 130
• --- hyperbolic, 87
• --- theorem, 138
• --- trigonometric, 76
• secant-tangent theorem, 138
• section, golden, 2
4
• 4
• 193
• 845
3
• sector formula, 666
• sector, spherical, 157
• segment
• --- normal, 227
• --- notion, 128
• --- polar normal, 227
• --- polar tangent, 227
• --- tangent, 227
• selection, 870
• self-similar, 823
• semantically equivalent, 287
• --- expressions, 345
• semifocal chord
• --- ellipse, 198
• --- hyperbola, 200
• --- parabola, 69
4
• 203
3
• semigroup. 299
• --- free, 299
• semimonotone, 613
• seminorm, 624
• semiorbit, 797
• sense class, 133
• --- of a figure, 132
• sensitive, with respect to the initial values, 829
• sentence, 286
• sentence-forming functor, 286
• sentential operation, 286
• separable sets, 625
• separation
• --- constant, 540
• --- theorems (convex sets), 625
• --- variables, 489
4
• 524
• 540
3
• separatrix
• --- loop, 808
4
• 840
3
• surface, 806
4
• 812
3
• sequence, 404
• --- bounded, 598
• --- Cauchy sequence, 607
• --- convergente, 606
• --- finite, 597
• --- in metric space, 606
• --- infinite, 404
• --- numbers, 404
• --- bounded, 404
4
• 598
3
• --- bounded above, 405
• --- bounded below, 405
• --- convergence, 405
• --- converging to zero, 598
• --- divergence, 405
• --- finite, 597
• --- law of formation, 404
• --- limit, 405
• --- monotone, 404
• --- term, 404
• series, 18
4
• 404
3
• --- alternating, 410
• --- arithmetic, 18
• --- Banach space, 612
• --- Clebsch-Gordan, 306
• --- comparison criterion, 407
• --- constant term, 406
• --- convergence, 408
• --- absolute , 416
• --- non-uniform, 414
• --- uniform, 414
4
• 416
3
• --- convergence theorems, 406
• --- convergent, 406
• --- D'Alembert's ratio test, 407
• --- definite, 18
• --- divergence, 409
• --- divergent, 406
• --- expansion, 417
• --- Fourier, 420
• --- Laplace transformation, 719
• --- Laurent, 692
• --- Maclaurin, 418
• --- power series. 416
• --- Taylor, 389
4
• 417
• 691
3
• --- finite, 18
• --- Fourier, 421
• --- complex representation, 421
• --- function, 414
• --- domain of convergence, 414
• --- general term, 406
• --- geometric, 19
• --- infinite, 19
4
• 406
3
• --- harmonic, 406
• --- hypergeometric, 513
• --- infinite, 404
4
• 406
3
• --- integral test of Cauchy, 408
• --- Laurent, 692
• --- Maclaurin, 418
• --- Neumann, 568
• --- Neumann series, 620
• --- partial sum, 406
• --- power, 416
• --- expansion, 417
• --- inverse, 417
• --- remainder, 406
4
• 414
3
• --- root test of Cauchy, 408
• sum, 406
• --- Taylor, 389
4
• 417
3
• --- analytic function, 691
• --- uniformly convergent
• --- continuity, 415
• --- differentiation, 415
• --- integration, 415
• --- properties, 415
• --- Weierstrass criterion, uniform convergence, 415
• set, 290
• --- absorbing, 799
• --- algebra, fundamental laws, 292
• --- axioms of closed sets, 606
• --- axioms of open sets, 606
• --- Borel, 636
• set (continued)
• --- bounded in metric space. 606
• --- cardinality, 298
• --- closed, 606
• --- closure, 607
• --- compact, 628
4
• 800
3
• --- complex numbers, 34
• --- convex, 599
• --- dense, 607
• --- denumerable infinite, 298
• --- disjoint, 292
• --- element, 290
• --- empty, 291
• --- equality, 291
• --- equinumerous, 298
• --- fundamental 292
4
• 616
3
• --- fuzzy, 360
• --- image, 48
• --- infinite, 298
• --- integers, 1
• --- invariant, 799
• --- chaotic, 828
• --- fractal, 828
• stable, 799
• --- irrational numbers, 2
• --- linear, 597
• --- manyness, 298
• --- measurable, 635
• --- natural numbers, 1
• --- non-denumerable infinite, 298
• --- notion of set, 290
• --- open, 605
• --- operation
• --- Cartesian product, 294
• --- complement, 292
• --- difference, 293
• --- intersection, 292
• symmetric difference, 293
• --- union, 292
• --- operations, 291
• --- order-bounded, 602
• --- original space, 48
• --- power, 298
• --- power set, 291
• --- quotient set, 297
• --- rational numbers, 1
• --- real numbers, 2
• --- relative compact, 628
• subset, 291
• --- theory, 290
• --- universal, 292
• --- void, 291
• sets, 290
• --- coordinates x,y. 294
• --- difference, symmetric, 293
• sexagesimal degree, 130
• shift mapping, 820
• shift register, lineare, 316
2
• shooting method, --- process,
3
• shooting me
• simple, 910
• shore-to-ship bearing, 171
• SI system, 1012
• side or leg of an angle, 128
• side-condition, 555
• Sierpinski
• --- carpet, 824
• --- gasket, 824
• sieve of Eratosthenes, 320
• --- (T-additivity, 635
• --- (T-algebra, 635
• --- Borelian, 636
• sign of a function, 49
• signal, 740
• --- analysis, 740
• synthesis, 740
• signature, universal algebra, 340
• significance, 779
• --- level, 753
4
• 776
3
• similarity transformation, 280
4
• 281
3
• simplex
• --- method, 850
4
• 851
3
• --- revised, 855
• step, revised, 856
• --- tableau, 851
• --- revised, 855
• Simpson's formula, 900
• simulation
• --- digital, 783
• --- Monte Carlo, 783
• sine
• --- geometric definition, 130
• --- hyperbolic, 87
• --- geometric definition, 131
• --- trigonometric, 74
• sine integral, 460
4
• 696
3
• sine law, 142
4
• 164
3
• sine-cosine law, 164
• --- polar, 164
• sine-Gordon equation, 547
4
• 549
3
• single-target method, 910
• singleton, 363
• singular value, 285
4
• 820
3
• --- decomposition, 285
• singular value decomposition, 285
• singularity
• --- analytic function, 673
• --- essential, 673
4
• 693
3
• --- isolated, 692
• --- pole, 693
• --- removable, 673
• sink, 804
4
• 811
3
• --- vector field, 653
• --- vertex, 358
• sinusoidal amount, 82
• slack variable, 847
• Slater condition, 863
• slide rule, 10
• --- logarithmic scale, 114
• slope
• --- plane, 194
• --- tangent, 227
• small circle, 158
4
• 175
3
• --- arclength, 175
• --- course angle, 176
• --- intersection point, 176
• --- radius, plane, 175
• --- radius, spherical, 17o
• smoothing
• --- continuous problem, 919
• --- parameter. 933
• spline, 932
• --- cubic, 932
• Sobolev space, 613
• solenoid, 828
• solid angle, 151
• soliton
• --- antikink, 550
• --- antisoliton, 549
• --- Boussinesq, 551
• --- Burgers, 551
• --- differential equation, partial, non-linear, 546
• --- Hirota, 551
• --- interaction, 547
• --- Kadomzev-Pedviashwili, 551
• --- kink, 550
• --- lattice, 550
• --- kink-antikink. 550
• --- collision, 550
• --- doublet, 550
• --- kink-kink
• --- collision, 550
• --- Korteweg de Vries, 548
• --- non-linear, Schrodinger. 549
• solution
• --- point, 847
• --- product form, method separation of variables, 524
• SOR method (successive overrelaxation), 896
• source, 804
4
• 811
3
• --- distribution
• --- continuous. 669
• --- discrete, 669
• --- field
• --- irrotational, 667
• --- pure, 667
• --- vector field, 653
• --- vertex, 358
• space
• --- abstract, 48
• --- Banach, 612
• --- complete, metric, 607
4
• 608
3
• --- directions, 210
• --- finite-dimensional, 628
• --- higher-dimensional, 117
• --- Hilbert, 615
• --- infinite-dimensional, 599
• --- isometric, 611
• --- Kantorovich, 603
4
• 628
3
• --- linear, 596
• --- p space, 639
• --- metric, 604
• --- completion, 610
• --- convergence of sequence, 606
• --- normable, 612
• separable, 607
• --- non-reflexive, 626
• --- normed
• --- axiom, 611
• --- properties, 612
• --- ordered normed, 613
• --- orthogonal, 616
• --- reflexive, 626
• --- Riesz, 602
• second adjoint, 626
• separable, 607
• Sobolev, 613
• --- unitary, 615
• --- vector, 596
• space curve, 238
• --- binormal, 239
4
• 241
3
• --- coordinate equation, 241
• --- curvature, 240
• --- direction, 238
• --- equation, 214
4
• 238
3
• --- moving trihedral, 239
• --- normal plane, 238
4
• 241
3
• --- osculating plane, 238
4
• 241
3
• --- principal normal, 239
4
• 241
3
• --- radius of curvature, 240
• --- radius of torsion, 242
• --- tangent, 238
4
• 241
3
• --- torsion, 242
• --- vector equation, 238
4
• 241
3
• space inversion, 269
• --- mixed product, 270
• scalar product, 270
• spectral radius, 620
• spectral theory, 622
• spectrum, 727
• --- amplitude, 727
• --- continuous, 623
• --- frequency, 727
• --- linear operator, 622
• --- phase, 727
• sphere, 156
• --- ellipsoid, 220
• --- equation, three forms, 243
• spherical
• --- biangle, 161
• --- coordinates, 209
• --- diagon,161
• --- distance, 158
• --- field, 643
• --- helix, 177
• --- lune, 161
• spiral, 103
• --- Archimedean, 103
• --- hyperbolic, 104
• --- logarithmic, 104
4
• 233
3
• spline
• --- basis spline, 933
• --- bicubic, 933
• --- approximation, 935
• --- interpolation, 933
• --- cubic, 931
• --- interpolation, 931
• smoothing, 932
• --- interpolation, 917
4
• 931
3
• --- natural, 931
• --- normalized Z?-spline, 933
• --- periodic, 931
• smoothing, 932
• --- matrix, 252
• --- tensor, 265
• square, 135
• stability
• --- absolutely stable, 907
• --- first approximation, 804
• --- integration of differential equation, 907
• --- Lyapunov stability, 803
• --- orbital, 803
• --- periodic orbit, 804
• --- perturbation of initial values, 907
• --- round-off error, numerical calculation, 942
• structural, 813
4
• 814
3
• standard
• --- deviation, 753
4
• 791
• 793
3
• --- error, 790
• --- normal distribution, 759
• state
• --- degenerate, 541
• --- particle, 538
• space, stochastic, 765
• stationary, 539
• statistics, 745
• --- descriptive, 772
• --- estimate, 771
• --- mathematical, 745
4
• 769
3
• sample function, 770
• steady state, 797
• Steffensen method, 886
• step
• --- from n to n + 1
4
• 5
3
• --- function, 697
4
• 714
• 734
• 744
3
• --- interval parameter, 897
• size, 904
• --- change, 905
• steradian, 151
• stereometry, 150
• Stieltjes
• --- integral, 628
• --- comparison with Riemann integral, 453
• --- notion, 453
• --- transformation, 707
• Stirling formula, 462
• stochastic
• --- basic notions, 765
• --- chains, 766
• --- process, 765
2
• processes, --- trigonometric,
3
• --- processes, 765
• stochastics, 745
• Stokes, integral theorem, 666
• strangling, 264
• Strategie
• --- mutations-selektion-, 871
• stream function, 681
• strip, characteristic, 519
• strophoide, 95
• structure
• --- algebraic, 286
• --- classical algebraic, 298
• Sturm
• --- chain, 44
• --- function, 44
• sequence, 889
• --- theorem, 44
• Sturm�Liouville problem, 515
• sub determinant, 259
• subdomain, 933
• subgraph, 349
• --- induced, 349
• subgroup, 300
• --- criterion, 300
• --- cyclic, 300
• --- invariant, 301
• --- normal, 301
• --- trivial, 300
• subinterval, 441
• subnormal, 227
• subring, 313
• --- trivial, 313
• subset, 291
4
• 597
3
• --- affne, 597
• --- linear, 597
• --- open,605
• subspace, 317
• --- affne, 597
• --- criterion, 317
• --- invariant, representation of groups, 305
• subtangent, 227
• subtraction
• --- complex numbers, 36
• --- computer calculation, 940
• --- polynomials, 11
• --- rational numbers, 1
• subtractive cancellation, 940
• sum, 6
• --- algebraic, 366
• --- drastic, 366
• --- of the digits
• --- alternating, first order, 322
• --- alternating, second order, 322
• --- alternating, third order, 322
• --- first order, 322
• second order, 322
• --- third order, 322
• --- residual squares, 921
• --- Riemann, 441
• --- rules of calculation, 6
• summation sign, 6
• --- transverse, 322
• --- vectors, 181
• summation convention, Einstein's, 262
• superposition
• --- fields, 669
• --- law, fields, 669
• --- non-linear, 549
• --- oscillations, 82
• --- principle, 683
• --- differential equation, linear, 507
• --- differential equations, higher order, 501
• supplementary angle formulas, 79
• support
• --- compact, 742
• --- measure, 816
• --- membership function, 360
• supporting functional, 625
• supremum, 602
• surface, 243
• --- area, double integral, 474
• --- B-B representation, 935
• --- barrel, 157
• --- block, 152
• --- cone, 156
• --- conical, 156
4
• 214
3
• --- constant curvature, 249
• --- cube, 152
• --- curvature of a curve, 247
4
• 249
3
• --- cylinder, 154
• --- developable, 250
• --- element, 247
• --- element, vector components, 660
• --- equation, 223
• --- in normal form, 220
• space, 213
• --- equation, space, 243
• --- first quadratic fundamental form, 245
• --- Gauss curvature, 249
• --- geodesic line, 250
• --- harmonic, 544
• --- integral, 479
4
• 663
• 664
3
• --- first type, 479
• --- general form, 485
• second type, 483
4
• 484
3
• --- line element, 245
• --- line of curvature, 248
• --- metric, 247
• --- minimal, 249
• --- normal, 244
4
• 246
3
• --- normal vector, 244
• --- oriented, 483
• --- patch, area, 247
• --- polyhedron, 151
• --- principal normal section, 248
• --- pyramid, 152
• --- quadratic, 223
• --- radius of principal curvature, 247
• --- rectangular parallelepiped, 152
• --- rectilinear generator, 222
• --- representation with splines, 931
• --- rotation symmetric, 213
• --- ruled, 250
• second order, 220
4
• 223
3
• --- central surfaces, 224
• --- invariant signs, 224
• --- types, 223
• second quadratic fundamental form, 248
• sphere, 156
• --- tangent plane, 244
4
• 246
3
• --- torus, 157
• --- transversal, 806
• surplus variable, linear programming, 850
• switch
• --- algebra, 342
4
• 346
3
• --- function, 346
• --- value, 346
• Sylvester, theorem of inertia, 282
• symbol
• --- internal representation (computer), 936
• --- Kronecker, 265
• --- Legendre, 329
• symmetry
• --- axial, 133
• --- central, 132
• --- element, 307
• --- Fourier expansion, 422
• --- group, 308
• --- applications in physics, 312
• --- crystallography, 310
• --- molecules, 308
• --- quantum mechanics, 312
• --- operation, 307
• --- crystal lattice structure, 310
• --- improper orthogonal mapping, 307
• --- reflection, 307
• --- rotation, 307
• --- without fixed point, 307
• --- with respect to a line, 133
• system
• --- chaotic according to Devaney, 829
• --- cognitive, 375
• --- complete, 618
• --- differential equation, higher order, 497
• --- linear, 505
• --- differential equations, partial
• --- canonical system, 519
• --- normal system, 519
• --- dynamical, 797
• --- chaotic, 829
• --- conservative, 798
• --- continuous, 797
• --- C''-smooth, 797
• --- discrete, 798
• --- dissipative, 798
• --- ergodic, 817
• --- invertible, 797
• --- mixing, 817
• --- motion, 797
• --- time continuous, 797
• --- time discrete, 797
• --- time dynamical, 798
• --- volume decreasing, 798
• --- volume preserving, 798
• --- volume shrinking, 798
• --- equation
• --- numerical solution, 890
• --- four points, 213
• --- generators, 300
• --- knowledge based interpolation, 377
• --- linear equations, 271
4
• 272
3
• --- compatible, 273
• --- consistent, 273
• --- fundamental system, 273
• --- homogeneous, 272
• --- inconsistent, 273
• --- inhomogeneous, 272
• --- overdetermined, 277
• --- pivoting, 274
• --- row echelon form, 276
• solvable, 273
• --- trivial solution, 273
• --- mixing, 829
• --- Morse-Smale, 815
• system (continued)
• --- normal equations, 277
• --- orthogonal, 616
• --- orthonormal, 617
• --- term-substitutions, 340
• --- trigonometric, 616
2
1
• T
2
• T
• table with double entry, --- element of curved surface,
3
• table with double e
• tacnode, curve, 232
• tangent
• --- circle, 198
• --- formula, 142
• --- geometric definition, 130
• --- hyperbolic, 87
• --- geometric definition. 131
• --- law, 142
• --- plane, 158
4
• 395
3
• --- surface, 244
4
• 246
3
• --- plane curve, 226
• --- polygon, 137
• --- space curve, 238
4
• 241
3
• trigonometric, 76
• tautology
• --- Boolean function, 344
• --- predicate logic, 290
• --- propositional logic, 288
• Taylor
• --- expansion, 55
4
• 389
• 417
3
• --- analytic function, 691
• --- one variable, 417
• --- several variables, 397
• --- vector function, 643
• --- formula, 389
• --- several variables, 397
• two variables, 396
• --- series, 389
4
• 417
3
• --- analytic function, 691
• --- one variable, 417
• --- several variables, 397
• theorem, 389
• --- several variables, 396
• telegraphic equation, 531
• telephone-call distribution, 769
• tensor, 262
• --- addition, subtraction, 268
• --- alternating, 265
• --- antisymmetric, 264
• --- components, 262
• --- contraction, 264
4
• 268
3
• --- definition, 262
• --- dyadic product, 264
• --- eigenvalue, 264
• --- generalized Kronecker delta, 265
• --- inertia, 263
• --- invariant, 265
• --- multiplication, 268
• --- oversliding, 268
• --- product, 263
• --- vectors, 255
• --- rank n, 262
• --- rank 0
4
• 263
3
• --- rank 1
4
• 263
3
• --- rank 2
4
• 263
3
• --- rules of calculation, 263
4
• 264
3
• --- skew-symmetric, 264
4
• 268
3
• --- spur, 265
• --- symmetric, 264
4
• 268
3
• tension, 263
• trace, 265
• tensor product approach, 934
• tent mapping, 816
• term algebra, 341
• test problem, linear, 907
• test, Kasiski-Friedman, 337
• test, statistical, 774
• tetraeder, 213
• tetragon, 136
• tetrahedron, 153
• Thales theorem, 139
4
• 141
3
• theorem
• --- Abel, 416
• --- Afraimovich-Shilnikov, 842
• --- alternating point, 923
• --- Andronov-Pontryagin, 814
• --- Andronov-Witt, 804
4
• 805
3
• --- Apollonius, 199
• --- Arzeli-Ascoli, 628
• --- Baire category, 608
• --- Banach
• --- continuity, inverse operator, 621
• --- fixed-point, 631
• --- Banach-Steinhaus, 620
• --- Bayes, 750
4
• 751
3
• --- Berge, 356
• --- binomial. 12
• --- Birkhoff, 342
4
• 817
3
• --- Block-Guckenheimer-Misiuriewicz, 829
• --- Bolzano
• --- one variable, 59
• --- several variables, 123
• --- boundedness of a function
• --- one variable, 60
• --- several variables, 123
• --- Cauchy integral theorem, 688
• --- Cayley, 302
4
• 355
3
• --- center manifold
• --- differential equations, 830
• --- mappings, 835
• --- Chebyshev, 436
• --- Chinese remainder, 328
• --- Clebsch-Gordan, 306
• --- closed graph, 620
• --- constant analytic function, 673
• --- convergence, measurable function, 638
• --- decomposition, 297
• --- Denjoy, 844
• --- differentiability, respect to initial conditions, 797
• --- Dirac, 353
• --- Douady-Oesterle, 826
• --- Euclidean
• --- algorithm, 324
• theorems. 141
• --- Euler, 154
4
• 331
3
• --- Euler-Hierholzer, 352
• --- Fatou, 638
• --- Fermat, 331
4
• 388
3
• theorem (continued I)
• --- Fermat-Euler, 331
• --- fixed point theorem
• --- Banach, 608
• --- fixed-point
• --- Banach, 631
• --- Brouwer, 633
• --- Schauder, 633
• --- Floquet, 803
• --- fundamental integral calculus, 458
• --- Girard, 164
• --- Grobman-Hartman, 811
4
• 813
3
• --- Hadamard-Perron, 807
4
• 812
3
• --- Hahn (extension theorem), 625
• --- Hellinger-Toeplitz, 620
• --- Hilbert-Schmidt, 630
• --- Holladay, 932
• --- Hurwitz, 503
• --- intermediate value
• --- one variable, 59
• --- several variables, 123
• --- KAM (Kolmogorov-Arnold-Moser), 815
• --- Krein-Losanovskij, 621
• --- Kuratowski, 357
• --- Lagrange, 301
• --- Lebesgue. majorized convergence, 638
• --- Ledrappier, 826
• --- Leibniz, 410
• --- Lerey-Schauder, 633
• --- LeviB., 638
• --- limits
• --- functions, 405
• --- sequences of numbers, 405
• --- Liouville, 673
4
• 801
3
• --- Lyapunov, 803
• --- maximum value, analytic function, 673
• --- Meusnier, 247
• --- nested balls, 608
• --- Ore, 353
• --- Oseledec, 820
• --- Palis-Smale, 815
• --- Picard-Lindelof, 610
4
• 797
3
• --- Poincare-Bendixson, 805
• --- Posa, 353
• --- Ptolemy's, 136
• --- Pythagoras
• --- general triangle, 142
• --- orthogonal space, 616
• --- right-angled triangle, 141
• --- Radon-Nikodym, 638
• --- Riemann, 409
• --- Riesz, 624
• --- Riesz-Fischer, 618
• --- Rolle, 388
• --- Schauder, 629
• --- Schwarz, exchange, 395
• --- Sharkovsky, 829
• --- Shilnikov, 840
• --- Shinai, 829
• --- Shoshitaishvili, 830
• --- Smale, 840
• --- stability in the first approximation, 804
• --- Sturm, 44
• theorem (continued II)
• --- superposition law, 669
• --- Sylvester, of inertia, 282
• Taylor, 389
• --- one variable, 417
• --- several variables. 396
• Thales, 139
4
• 141
3
• total probability, 750
• Tutte, 356
• --- variation of constants, 802
• --- Weierstrass, 414
4
• 607
3
• --- one variable, 60
• --- several variables. 123
• --- Wilson, 331
• --- Wintner-Conti, 800
• --- Young, 824
4
• 825
3
• theory
• --- distribution, 714
• --- elementary number, 320
• --- field, 669
• --- function, 671
• --- graph, algorithms, 348
• --- probability, 745
• --- risk, 21
• --- set, 290
• --- spectral, 622
• --- vector fields, 642
• theta function, 705
• time frequency analysis, 743
• tolerance, 364
• topological
• --- conjugate, 810
• --- equivalent, 810
• torsion, space curve, 242
• torus, 157
4
• 828
• 829
3
• --- differential equation, linear, autonomous, 803
• --- dissolving, 841
• --- formation, 837
4
• 842
3
• --- invariant set, 806
• --- losing smoothness, 842
• --- resonance torus, 842
• total curvature, 243
• trace
• --- matrix, 252
• tensor, 265
• tractrix, 106
• trail, 351
• --- Euler, 352
• --- open, 352
• trajectory, 797
• transform, 707
• transformation
• --- Cartesian into polar coordinates, 400
• --- covering, 299
• --- determinant, 211
• --- element of area, 474
• --- element of curved surface, 482
2
• geometrical, --- type, universal algebra,
3
• --- geometrical, 141
• --- Hopf-Cole, 551
• --- identical, 11
• --- invariance, 265
• --- linear, 262
4
• 319
• 600
3
• --- method, 283
• transformation (continued)
• --- orthogonal coordinates, 210
• --- principal axes, 264
4
• 281
3
• --- quadratic form, 281
• --- similarity, 280
4
• 281
3
• --- wavelet transformation, 740
• transition
• --- matrix, 766
4
• 767
3
• --- probability, stochastic, 766
• transitivity, 29
• translation
• --- invariance, 265
• --- primitive, 310
• transport, network, 358
• transportation problem. 857
• transposition law, 288
• trapezoid, 135
• --- Hermite's trapezoidal formula, 900
• trapezoidal
• --- formula, 899
• --- sum, 899
• traversing, 147
• tree, 354
• --- hight, 354
• --- ordered binary, 354
• --- regular binary, 354
• --- rooted, 354
• --- spanning, 354
4
• 355
3
• --- minimum, 355
• triangle
• --- altitude, 132
• --- area, 193
• --- bisector, 132
• --- center of gravity, 132
• --- circumcircle, 132
• --- congruent, 133
• --- coordinates, 917
• --- equilateral, 132
• --- Euler, 162
• --- incircle, 132
• --- inequality, 181
• --- axioms of norm, 258
• --- complex numbers, 30
• --- metric space, 604
• --- norm, 611
• --- real numbers, 30
• --- unitary space, 615
• --- isosceles, 132
• --- median, 132
• --- orthocenter, 132
• --- Pascal, 13
• --- plane. 131
• --- area, 141
4
• 143
3
• --- basic problems, 143
• --- Euclidean theorems, 141
• --- general, 141
• --- incircle, 143
• --- radius of circumcircle, 142
• --- right-angled, 132
4
• 141
3
• tangent formula, 142
• --- polar, 162
• --- similar, 133
• --- spherical, 161
• --- basic problems, 167
• --- calculation, 167
• --- Euler, 162
• --- oblique, 169
• --- right-angled, 168
• triangular
• --- decomposition, 891
• --- matrix, 253
• --- lower, 253
• --- upper, 253
• triangularization, FEM (finite element method), 914
• triangulation, geodesy
• --- first fundamental problem, 147
• --- second fundamental problem, 148
• trigonometry
• --- plane, 141
• --- spherical, 158
• trihedral
• --- angle, 151
4
• 163
3
• --- moving, 238
4
• 239
3
• triple (ordered 3
4
• 294
3
• triple integral, 476
• --- application, 480
• trochoid, 100
• truncation, measurement error, 787
• truth
• --- function, 286
4
• 287
3
• --- conjunction, 286
• --- disjunction, 286
• --- equivalence, 286
• --- implication, 286
• --- NAND function, 288
• --- negation, 286
• --- NOR function, 288
• table, 286
• --- value, 286
• turbulence, 798
4
• 841
3
• Tutte theorem, 356
• two lines, transformation, 205
• two-body problem, 520
• type, universal algebra, 340
2
1
• U
2
• U
• umbilical point, 249
• uncertainty
• --- absolute, 792
• --- fuzzy, 360
• --- relative, 792
• ungula, cylinder, 155
• union
• --- fuzzy sets, 365
3
• 366
2
• --- sets, 292
• units
• --- SI system, 1012
• units, physical, 1012
• universal quantifier, 289
• universal substitution, 438
• urn model, 755
1
• V
2
• V
• vagueness, --- cone,
3
• vaguene
• valence
• --- in-valence, 348
• --- out-valence, 348
• value
• --- expected, measurent, 753
• --- system (function of several variables), 117
• --- true, measurement. 790
• van der Pol differential equation, 833
• variable
• --- artificial, 854
• --- basic, 850
• --- Boolean, 344
• --- bound variable, predicate logic, 289
• --- dependent, 47
4
• 271
3
• --- free, predicate logic, 289
• --- independent, 47
4
• 117
• 271
3
• --- linguistic, 361
• --- non-basic, 850
• --- propositional, 286
• --- random, 751
• variance, 753
• --- distribution, 753
• --- sample function, 771
• --- statistics, 773
• --- two-dimensional distribution, 780
• variation
• --- function, 60
• --- of constants, method, 501
• variation of constants
• --- differential equation, linear, 507
• --- theorem, 802
• variational calculus, 552
• --- auxiliary curve, 554
• --- comparable curve, 554
• --- Euler differential equation, 554
• --- side-condition, 552
• variational equation, 804
4
• 821
• 914
3
• --- equilibrium point points, 811
• variational problem, 552
4
• 909
3
• --- brachistochrone problem, 553
• --- Dirichlet, 559
• --- extremal curves, 553
• --- first order, 552
• --- first variation, 561
• --- functional, 553
• --- higher order, 552
• --- higher order derivatives, 556
• --- isoperimetric, general, 553
• --- more general, 560
• --- numerical solution, 560
• --- direct method, 560
• --- finite element method, 561
• --- gradient method, 561
• --- Ritz method, 560
• --- parameter representation, 552
• --- parametric representation, 557
• --- second variation, 561
• --- several unknown functions, 557
• --- side-conditions, 555
• --- simple
• --- one variable, 553
• --- several variable, 558
• variety, 341
• vector
• --- absolute value, 180
• --- affhne coordinates, 182
• --- axial, 180
• --- reflection behavior, 268
• --- base, 186
• --- base vector
• --- reciprocal, 185
• --- bound,180
• --- Cartesian coordinates, 182
• --- column, 253
• --- components, 648
• --- conjugate, 869
• --- coordinates, 182
• --- Darboux vector, 243
• --- decomposition, 182
• --- diagram, oscillations, 83
• --- differentiation rules, 642
• --- direction in space, 180
• --- direction, vector triple, 180
• --- directional coefficient, 183
• --- expansion coefficient, 183
• --- fields, 642
• --- free, 180
• --- left singular, 285
• --- length, 189
• --- line, 648
• --- magnitude, 180
• --- matrix, 253
• --- metric coefficients, 185
• --- notion, 180
• --- null vector, 181
• --- polar, 180
• --- reflection behavior, 268
• --- pole, origine, 181
• --- position vector, 181
• --- complex-number plane, 34
• --- radius vector, 181
• --- complex-number plane, 34
• --- reciprocal, 186
• --- reciprocal basis vectors, 186
• --- residual, 277
• --- right singular, 285
• --- row, 253
• --- scalar invariant, 657
• --- sliding, 180
• --- space, 597
• --- stochastic, 766
• --- unit, 180
• --- zero vector, 181
• vector algebra, 180
• --- geometric application, 189
• --- notions and principles, 180
• vector analysis, 642
• vector equation
• --- line, 188
• --- plane, 188
• --- space curve, 238
4
• 241
3
• vector equations, 187
• vector field, 645
4
• 797
3
• --- Cartesian coordinates, 646
• --- central, 645
• --- circular field, 646
• --- components, 648
• --- contour integral, 662
• --- coordinate definition, 646
• vector field (continued)
• --- cylindrical, 646
• --- cylindrical coordinates, 647
• --- directional derivative, 650
• --- divergence, 653
• --- point-like source, 668
• --- rotation, 654
• --- sink, 653
• --- source, 653
• --- spherical, 646
• --- spherical coordinates, 647
• vector function, 642
4
• 645
3
• --- derivative, 642
• --- differentiation, 642
• --- hodograph, 642
• --- linear, 267
• --- scalar variable, 642
• --- Taylor expansion, 643
• vector gradient, 652
• vector iteration, 283
4
• 284
3
• vector lattice, 602
• --- homomorphism, 603
• vector potential, 668
• vector product, 183
• --- hints, 255
• --- representation in coordinates, 187
• vector space, 316
4
• 596
3
• --- all null sequences, 598
• --- bounded sequences, 598
• --- B(T), 598
• --- (,'([a,b]), 598
• --- e(v([a,b]),rm
• --- complex, 597
• --- convergent sequences, 598
• --- Euclidean, 318
• --- finite sequence of numbers, 597
• --- F', 597
• --- T(T), 598
• --- functions, 598
• --- infinite-dimensional, 317
4
• 599
3
• --- L p, 639
• --- p, 598
• --- n-dimensional, 317
• --- n-dimensional Euclidean, 254
• --- ordered by a cone, 601
• --- partial ordering, 601
• --- real, 316
4
• 597
3
• --- s of all sequences, 597
• --- sequences, 597
• vector subspace
• --- stable, 807
4
• 812
3
• --- unstable, 807
4
• 812
3
• vectors, 180
4
• 253
3
• --- angle between, 189
• --- collinear, 181
• --- collinearity, 184
• --- commutative law, 255
• --- coplanar, 181
• --- cyclic permutation, 207
• --- double vector product, 184
• --- dyadic product, 255
• --- equality, 180
• --- Lagrange identity, 185
• --- linear combination, 181
4
• 184
3
• --- mixed product, 184
4
• 187
3
• --- Cartesian coordinates, 185
• --- orthogonality, 184
• --- products
• --- affine coordinates, 185
• --- Cartesian coordinates, 185
• --- products, properties, 183
• --- scalar product, 255
• --- Cartesian coordinates, 185
• --- representation in coordinates, 186
• --- sum, 181
• --- tensor product, 255
• --- triple product, 184
• vector product
• --- representation in coordinates, 187
• Venn diagram, 291
• verification, proof, 5
• vertex
• --- angle, 128
• --- degree, 348
• --- ellipse, 198
• --- graph,348
• --- hyperbola, 200
• --- initial, 348
• --- isolated, 348
• --- level, 354
• --- parabola, 203
• --- plane curve, 232
• --- sink, 358
• --- source, 358
• --- terminal, 348
• vertical angle, 143
• vertices, distance, 351
• vibration, differential equation
• --- bar, 526
• --- round membrane, 527
• --- string, 525
• Vieta, root theorem, 43
• Volterra integral equation, 563
4
• 609
3
• --- first kind, 585
• --- second kind, 587
• volume
• --- barrel, 157
• --- block, 152
• --- cone, 156
2
• cube, --- volume scale,
3
• --- cube, 152
• --- cylinder, 154
• --- double integral, 474
• --- element, vector components, 660
• --- hollow cylinder, 155
• --- obelisk, 153
• --- parallelepipedon with vectors, 189
• --- polyhedron, 151
• --- prism, 152
• --- pyramid, 152
• --- rectangular parallelepiped, 152
• --- sphere, 156
• --- subset, 635
• --- tetraeder, 213
• --- torus, 157
• --- triple integral, 480
• --- wedge, 154
• volume derivative. 651
• volume integral, 476
• volume scale, 114
2
1
• W
2
• W
• Walsh
• --- functions, 744
• --- systems, 744
• wave
• --- amplitude, 82
• --- frequency, 82
• --- length, 82
• --- period, 82
• --- phase, 82
• --- plane, 741
• wave equation
• -dimensional, 536
• --- one-dimensional, 732
• --- Schrodinger equation, 539
• wave function
• --- classical problem, 536
• --- heat-conduction equation, 537
• --- Schrodinger equation, 538
• wavelet, 741
• --- Daubechies, 742
• --- Haar, 741
• --- Mexican hat, 741
• --- orthogonal, 742
• --- transformation, 740
3
• 741
2
• --- discrete, 743
• --- discrete, Haar, 743
• --- dyadic, 742
• --- fast, 743
• Weber
• --- differential equation, 545
• --- function, 509
• wedge, 154
• Weibull distribution, 761
• Weierstrass
• --- approximation theorem, 607
• --- criterion uniform convergence, 415
• --- function, 705
• --- theorem, 414
• --- one variable, 60
• --- several variables, 123
• weight
• --- measurement, 793
• --- of orthogonality, 515
• --- statistical, 753
• weighting factor, statistical, 790
• witch of Agnesi, 94
• word, 604
• work (mechanics)
• --- general, 469
• --- special, 451
• Wronskian determinant, 500
3
• 802
2
• --- zero-point vibration energy, 546
• --- zeropoint, 1
• --- z-transformation, 733
• --- convolution, 736
• --- damping, 736
• --- definition, 734
• --- difference, 735
• --- differentiation, 736
• --- integration, 736
• --- inverse, 737
• --- original sequence, 734
• --- rules of calculation, 735
• --- summation, 735
• --- transform, 734
• --- translation, 735
• --- z-transformable, 734
1
• X
2
• X
• empty
1
• Y
2
• Y
• Young scheme, 306
1
• Z
2
• Z
• zenith, 143
• --- distance, 143
• zero matrix, 251
• zero-point translational energy, 541
1
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