D
d'Alembert formula, 536 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, 913 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, 640 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, 317 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, 395 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, 802 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, 740 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, 395 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, 501 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 fraction of linear functions, 494 general solution, 487 general, n-th order, 487 differential equation, ordinary (continued) graphical solution, 497 homogeneous, 489 implicit, 487, 492 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, 811 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, 672 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, 841 heat conduction equation one dimensional, 531 one-dimensional, 529 three-dimensional, 537 Helmholtz, 540 hyperbolic type, 522, 524 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, 667, 670 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, 658 gradient, 650, 651, 658 Laplace, 657, 658 nabla, 656, 658 non-linear, 632 relations, 658 rotation, 654, 658 rules of calculations, 658 space, 649, 656 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, 395 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, 599 Dirac distribution, 640 measure, 636, 816 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, 670 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, 640, 717 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, 756 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, 323 division 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, 687 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