Handbook of mathematics for engineers and scienteists part 214 ppt

INDEX 1459 condition (continued) Lax, 792 Legendre, 998, 999, 1004, 1007 Legendre, strengthened, 999, 1004, 1007 line and plane to be parallel, 140 line and plane to be perpendicular, 140 line to be entirely contained in plane, 140 Lipschitz, 248, 310, 313, 543 necessary of extremum, 269 planes to be parallel, 138 planes to be perpendicular, 138 planes to coincide, 138 stability for generalized solution, 568 straight lines to be parallel on plane, 96 straight lines to be perpendicular on plane, 95 straight lines to coincide on plane, 96 three points to be collinear, 93 transversality, 577 two lines in space to meet, 143 two lines to be parallel in space, 135 two lines to be perpendicular in space, 136 two planes to be parallel, 138 two planes to be perpendicular, 138 unitarity, 206 vectors to be parallel, 122 vectors to be perpendicular, 122 Weierstrass, 998 conditional convergence, 342 conditional cumulative distribution function, 1062 conditional distribution, 1062 conditional expectation, 1063, 1066 conditional extremum of function, 271 conditional minimum, 270 conditional probability, 1035 conditionally convergent series, 342 conditions Cauchy–Riemann, 402 compatibility for PDEs, 738, 740, 742, 744 compatibility for systems of linear equations, 757 compatibility of initial and boundary conditions, 607, 609 d’Alembert–Euler, 402 Dirichlet, 357 evolutionary, 792 existence of extremum, 258 extremum of function of two variables, 269 radiation, 1290 Sommerfeld, 1290, 1291 sufficient of extremum, 269 transversality, 1001 Weierstrass–Erdmann, 997, 999 cone, 67, 68, 145 asymptotic, 144 frustum, 68 right circular, 68, 145 confidence boundary, 1081 confidence interval, 1093 confidence level, 1093 conformal mapping, 419 congruence transformation, 182 congruent matrices, 182 congruent polyhedral angles, 61 congruent transformation, 182 conic point, 391 of surface, 391 conical coordinates, 1202 conical surface, 67 conjugate complex number, 399 conjugate diameters, 151 of ellipse, 100 of hyperbola, 104 conjugate element of group, 231 conjugate matrix, 170 conjugate point, 998, 1000, 1004, 1007 conjunctive matrices, 183 conjunctive transformation, 183 conservation law, 564, 766, 996 at discontinuity, 561 for PDEs, 766 consistency condition for general linear system, 198 consistency of estimator, 1089 consistent system of linear algebraic equations, 197 constant Euler, xxx, 944, 949, 958 H ¨ older, 310 separation, 678 constrained maximum, 270 constrained minimum, 270 constraint, 270 differential, 737, 738, 1009, 1010 differential, first-order, 739 differential, secod-order, 744 isoperimetric, 1002 contact transformation for ordinary differential equations, 468 for partial differential equations, 660 continuation, direct analytic, 418 continuity equation, 784 continuity theorem for characteristic functions, 1067 continuous bivariate random variable, 1059 continuous curve in space, 123 continuous distributions, 1051 continuous function, 243 of bounded variation, 249 continuous point group, 730 continuous probability distribution, mode, 1045 continuous random variable, 1040, 1064 expectation, 1042 continuous subgroups, 228 continuous surface in space, 123 contraction ratio, 403 control, 1010 controlled process, admissible, 1010 convergence almost sure, 1068 in distribution, 1069 in mean, 1068 in probability, 1068 mean-square, 315 1460 INDEX convergence (continued) nonuniform, 249 pointwise, 249 uniform, 249 weak, 1069 convergence criteria arbitrary series, 341 infinite products, 347 series with positive terms, 338 convergence criterion Abel, 341 Abel–Dini, 341 Bertrand, 340 Bugaev, 341 D’Alembert, 338 Dini, 341 Dirichlet, 342 Dirichlet–Jordan, 358, 361 Ermakov, 340 Gauss, 339 generalized Ermakov, 340 Kummer, 340 Lipschitz, 358 Lobachevsky, 341 Maclaurin–Cauchy, 339 Raabe, 339 Sapogov, 340 Weierstrass, 349 convergence domain of series, 348 convergence radius of power series, 350, 411 convergence theorem, collocation method, 524 convergence with probability 1, 1068 convergent infinite products, 346 convergent sequence, 237, 238 convergent series, 337, 338 convex function, 245 convex polygon, 51 convex polyhedron, 61 convex programming, 1027 convolution theorem, 437, 444 Fourier transform, 444 Laplace transform, 437 convolution-type equation, 825, 829 coordinate axes in space, 114 on plane, 78 coordinate functions, 847, 851 coordinate net, 387 coordinate of point on axis, 77 coordinate space, 188 coordinate system affine, 190 in space, 114 left rectangular Cartesian, 79 oblique Cartesian, 79 on axis, 77 on plane, 78 polar, 80 rectangular Cartesian, 78, 114 right rectangular Cartesian, 79 coordinate vectors, 388 coordinates bicylindrical, 326, 1204 bipolar, 1204 Cartesian, in plane, 79 Cartesian, in space, 115 center of mass of body, 327 center of mass of flat plate, 324 conical, 1202 current, 84 curvilinear, 387 cylindrical, xxxi, 118, 326, 1198 element, 188, 189 ellipse canonical, 98 elliptic, 1202 elliptic cylinder, 1202 first prolongation, 718 Gaussian, 387 general orthogonal curvilinear, 1196 hyperbola canonical, 101 line, 77 oblate ellipsoid of revolution, 1200 orthogonal, 1196 parabola canonical, 104 parabolic, 1203 parabolic cylinder, 1203 plane, 78 point, 79, 115 polar, 80, 1198 second prolongation, 718 spherical, xxxi, 119, 326, 1199 toroidal, 326, 1205 coplanar vectors, 113 correlated random variables, 1061 correlation function, 1072 matrix, 1066 random variables, 1061 sample, 1104 two random variables, 1061 correspondence, isometric, 395 cosecant, xxx, 24 coset, 227 left, 227 right, 227 cosine, 24, 25 cosine integral, 942 cost, reduced, 1015 cotangent, xxx, 24, 26 countable set, 3 counterexample, 17 covariance, 1060, 1061 matrix, 1065 of two random variables, 1061 Cramer rule, 199 Cram ´ er–Rao efficient unbiased estimator, 1089 Cram ´ er–Rao inequality, 1089 criterion Cauchy convergence for series, 339 Dini convergence for series, 341 Dirichlet convergence, 342, 350 Dirichlet–Jordan convergence for Fourier series, 358, 361 INDEX 1461 criterion (continued) Ermakov convergence for series, 340 first comparison, 338 Gauss convergence for series, 339 Kummer convergence for series, 340 Leibnitz convergence for series, 341 Lipschitz convergence for series, 358, 361 Lobachevsky convergence for series, 341 Maclaurin–Cauchy convergence for series, 339 Routh–Hurwitz, 167 Sapogov convergence for series, 340 second convergence, 338 special Cauchy, 341 Sylvester, 219 Weierstrass convergence, 349 critical region, 1095 cross-correlation function, 1072 cross-covariance function, 1072 cross product of vectors, 121 cube, 62 cubic equation, 158 cumulative distribution function, 1039 conditional, 1062 cumulative frequency, 1084 cumulative relative frequency, 1084 curl, 331 of vector fi eld, 331 334 curvature, 376, 384 center, 375 circle, 375 extrinsic, 394 Gaussian, 394 geodesic, 392 lines, 393 mean, 394 normal, 392, 394 of curve on surface, 392 of plane curves, 376 of space curves, 384 radius, 376, 384 vector, 383 curve discriminant, 378 in space, 120, 123 intrinsic equations, 388 on plane, 84 parabolic type of second-order, 108 second-order, 107 in space, 119, 123 on plane, 84 on plane algebraic, 87 curves, parametric, 387 curvilinear coordinates, 387 on surface, 387 curvilinear integral, 329 curvilinear lines, 387 cusp, first kind, 372 cusp, second kind, 372 cycle, 1022 cyclic functional equation, 920 cyclic groups, 229 cylinder, 65, 66 base, 65 central, 225 function, 947 paraboloidal, 225 right, 66 right circular, 66 round, 66 segment, 67 truncated, 66 cylindrical coordinates, xxxi, 118, 1198 cylindrical function, 947, 1214 cylindrical Korteweg–de Vries equation, 1328 cylindrical surface, 65 cylindrical tube, 67 D D’Alembert convergence criterion for series, 338 D’Alembert equation, 1443 D’Alembert formula, 1278 D’Alembert (Gauss) formulas, 73 D’Alembert method, 533 D’Alembert solution for wave equation, 600 D’Alembert–Euler conditions, 402 damped vibrations, 1213 Darboux equation, 458 Darboux vector, 386 de Moivre formula, 30, 400 de Morgan laws, 1032 death process, 1077 decadic logarithm, 9 decile, 1045 decomposition of matrices, 173, 201 theorems, 208, 1055 triangular, 180 decreasing function, 236 decreasing sequence, 238 defect of matrix, 178 definite integral, 286, 1147 geometric applications, 299 geometric meaning, 287 physical applications, 299 properties, 287 definite integrals involving exponential functions, 1150 involving hyperbolic functions, 1152 involving logarithmic functions, 1152 involving trigonometric functions, 1153 degenerate bilinear form, 215 degenerate hypergeometric equation, 1215 degenerate hypergeometric function, xxxii, 956 Wronskian, 958 degenerate kernel, 801, 802, 810, 811, 825, 829 degenerate matrix, 178 degenerate quadratic form, 216 density, spectral, 1075 density multiplication theorem, 1062 dependent variable, 236 1462 INDEX derivation formulas, 396 derivative, 250 basic elementary functions, 252 directional, 267, 268 geometrical meaning, 251 higher-order, 255 of function, 402 of polynomial, 157 one-sided, 252 partial, 264 physical meaning, 251 total, 265 Descartes, Folium, 374 Descartes theorem, 165 Descartes–Euler solution, 160 descending Landen transformation, 974 determinant, 175 Casoratti, 901 column expansion, 176 Gram, 193 Jacobian, 321 linear operator, 209 order n, 176 order 2, 175 order 3, 176 product of matrices, 179 properties, 176 row expansion, 176 sum of matrices, 179 Vandermonde, 179 Wronskian, 473, 482, 518, 539 determined system of linear algebraic equations, 197 deviation of point from line, 93 of point from plane, 141 residual standard, 1063 root-mean-square, 1045 sample, 1088 standard, 1045 diagonal main, 168 secondary, 168 diagonal matrix, 168 diameter, 151 conjugate to chords, 108 conjugate to family of planes, 151 of circle, 56 of ellipse, 100 of hyperbola, 104 of parabola, 106 of second-order curve, 108 of set, 317 of sphere, 69 partition, 317 principal, of ellipse, 100 principal, of hyperbola, 104 principal, of parabola, 106 diameters conjugate, 151 conjugate, of ellipse, 100 conjugate, of hyperbola, 104 Dido problem, 1003 diet problem, 1012 difference common, 11 finite, 884 of complex numbers, 399 of events, 1031 of matrices, 169 of polynomials, 155 of vectors, 113, 188 difference equation, 889, 1409, 1422, 1425–1426 characteristic, 881 general solution, 874, 884 in integer argument, 873 linear, constant-coefficient, 873, 877 linear, higher-order, 1425 linear, second-order, 877, 1421 linear homogeneous, 881, 885, 889, 906, 904 linear homogeneous, first-order, 873 linear homogeneous, second-order, 877, 1421 linear nonhomogeneous, 883, 890–893 linear nonhomogeneous, constant-coefficient, first-order, 1411 linear nonhomogeneous, constant-coefficient, nth-order, 1427 linear nonhomogeneous, constant-coefficient, second-order, 1422 linear nonhomogeneous, first-order, 873 linear nonhomogeneous, nth-order, 1426 linear nonhomogeneous, second-order, 879, 880, 1421 linearly independent solutions, 882 logistic, 875 nonlinear, 884 nonlinear, first-order, 873 Riccati, 875, 918 difference kernel, 801, 804, 810, 813, 825, 829, 841 differentiable function, 251 differential, 250, 251, 265 higher-order, 255 differential constraint, 737, 1009, 1010 first-order, 739 second-order, 744 differential constraints method, 737 differential equation Abel, of first kind, 462 Abel, of first kind, canonical form, 463 Abel, of second kind, 464 Abel, of second kind, canonical form, 465 Airy, 955 autonomous, 1207, 1223, 1225 axisymmetric steady hydrodynamic boundary layer, 712 Bernoulli, 458, 1207 Bessel, 1216 Bessel, modified, 1216 biharmonic, 1297 biharmonic, homogeneous, 1297 biharmonic, nonhomogeneous, 1298 Born–Infeld, 659 INDEX 1463 differential equation (continued) Boussinesq, 709, 1332 Burgers, 664, 666, 751, 1307 Burgers, radial symmetric case, 1307 Burgers–Korteweg–de Vries, generalized, 709 characteristic, for system of ODEs, 529 Clairaut, 467, 572, 1262 Darboux, 458 degenerate hypergeometric, 1215 diffusion, nonlinear, with cubic source, 753 diffusion boundary layer, 1276 Duffing, 501 elliptic, 585, 588, 590, 1284 Emden–Fowler, 1223 Ermakov, 1222 Euler, 473, 520, 994, 998, 1216 evolution, second-order, 739 exact, first-order, 458 exact, second-order, 491 Fisher, 1299 Fitzhugh–Nagumo, 1302 free oscillations, 1213 gas dynamics, 558 Gaussian hypergeometric, 960, 1218 generalized homogeneous, 526 Hamilton–Jacobi, 576 heat, linear, 585, 1267 heat, linear, nonhomogeneous, 1268 heat, nonlinear, 1318 heat, stationary anisotropic, 1323 heat, stationary, with nonlinear source, 1320 heat, with axial symmetry, 1270 heat, with central symmetry, 1272 Helmholtz, 1289 Helmholtz, three-dimensional, 600 Helmholtz, two-dimensional, 599 homogeneous, 526, 591, 592, 810, 829, 834, 1207, 1224 homogeneous, generalized, 1210, 1224 hyperbolic, 585, 587, 590 hyperbolic, first canonical form, 587 hyperbolic, normal, 590 hyperbolic, second canonical form, 587 hypergeometric, 960, 1218 hypergeometric, degenerate, 1215 Jacobi, 998, 1004, 1007 Khokhlov–Zabolotskaya, 658 Khokhlov–Zabolotskaya, stationary, 1322 Klein–Gordon, 618, 625, 1280 Klein–Gordon, linear, 625 Klein–Gordon, nonlinear, 702, 1315 Klein–Gordon, two-dimensional, 618 Korteweg–de Vries, 752, 756, 759, 1327 Korteweg–de Vries, cylindrical, 1328 Korteweg–de Vries, modified, 1328 Lagrange, 467 Laplace, ODE, 520 Laplace, three-dimensional, PDE, 643 Laplace, two-dimensional, PDE, 585, 633, 1284 Legendre, 1217–1219 differential equation (continued) Lienard, 1225 light rays, 1259 linear, constant-coefficient, second-order, 473, 1213 linear homogeneous, second order, 1212 Liouville, 664 Mathieu, 980, 1221 Mathieu modified, 982, 1220 minimal surfaces, 770 model of gas dynamics, 1254 Monge–Amp ` ere, 1326 Monge–Amp ` ere, homogeneous, 668, 1326 Monge–Amp ` ere, nonhomogeneous, 1326 Newell–Whitehead, 733 ordinary, 453–550, 1207–1228 Painlev ´ e, 495 parabolic, 585, 586, 590 parabolic, in narrow sense, 590 partial, 553–798, 1247–1382 Pfaffian, 772, 773 Poisson, 1287 Poisson, three-dimensional, 599 Poisson, two-dimensional, 599 Riccati, 460 Riccati, general, 461, 462, 1209, 1210 Riccati, special, 1208 Schr ¨ odinger, linear, 1276 Schr ¨ odinger, nonlinear, 759, 1309 Schr ¨ odinger, of general form, 1311 Schr ¨ odinger, with cubic nonlinearity, 1309 Schr ¨ odinger, with power-law nonlinearity, 1310 separable, 456, 1207 separated, 456 sine-Gordon, 1314 sinh-Gordon, 1314 steady transonic gas flow, 1324 telegraph, 1284 telegraph, generalized, 626 thermal boundary layer, 1276 transonic gas flow, 1324 ultrahyperbolic, 590 vibration, with axial symmetry, 1282 vibration, with central symmetry, 1283 vibration of string, 1278 wave, 585, 1278 wave, nonlinear, 723, 1312–1317 wave, three-dimensional, 618 wave, two-dimensional, 618 Whittaker, 960 differential geometry, 367 differential operator linear, 591, 592 total, 740, 742 differential operators, of field theory, 272 differential substitution, 666 differentiation, 266 of implicit function, 266 of stochastic process, 1073 rules, 253 differentiation method for PDEs, 700 digamma function, 944, 949 1464 INDEX dihedral angle, 60 edge, 60 face, 60 dilatation ratio, 403 dimension of affine space, 190 of linear space, 189 of matrix, 167 of representation, 230 Dini convergence criterion for series, 341 Dini formula, 429 Dini integral, 1287 Diocles cissoid, 372 Dirac delta function, 599 direct analytic continuation, 418 direct method of symmetry reductions, 708 direct scattering problem, 760 direct sum of subspaces, 191 of two matrices, 175 directed segment, 77 direction, principal of second-order curve, 109 direction coefficients of straight line, 131 direction cosine of segment, 116 directional derivative, 267, 268 directrix, 65, 67 of ellipse, 99 of hyperbola, 102 of parabola, 105 Dirichlet conditions, 357 Dirichlet convergence criteria, 341, 342 Dirichlet criterion of uniform convergence of functional series, 350 Dirichlet problem, 427, 428 for elliptic equations, 593, 1330 Dirichlet theorem, 295, 358 representation of function by Fourier series, 357 Dirichlet–Jordan convergence criterion for Fourier series, 358, 361 Dirichlet–Mehler integral, 963 discontinuity, conservation law, 561 discontinuous solutions of quasilinear equations, 558 discrete bivariate random variable, 1058 discrete probability distribution, 1045 discrete probability space, 1033 discrete random variable, 1039, 1064 expectation, 1042, 1063 discrete subgroups, 228 discriminant large, 108 of algebraic equation, 163 small, 108 discriminant curve, 378 disk, 56, 97 dispersion equation, 532, 596 distance between elements, 193, 195 between lines in space, 142 between parallel lines, 96 distance (continued) between points on axis, 78 between points on plane, 81 between two parallel planes, 142 from point to line in space, 142 from point to plane, 141 from point to straight line, 93 shortest, 70 distribution bimodal, 1045 binomial, 1037, 1047 Cauchy, 1054 chi-square, 1054 conditional, 1062 continuous, 1051 convergence, 1069 exponential, 1052 Fisher–Snedecor, 1086 geometric, 1048 hypergeometric, 1049 Kolmogorov, 1086 log-normal, 1042 multimodal, 1045 negative binomial, 1050 normal, 1053 Poisson, 1050 Student, 1056 uniform, 1051 unimodal, 1045 distribution function empirical, 1084 joint, 1064 marginal, 1064 multivariate random variable, 1064 two-dimensional random vector, 1057 distribution polygon, 1085 divergence, 334 divergence theorem, 334 divergent infinite products, 346 divergent integral, 301 divergent series, 337 divisibility tests, 3 division of asymptotic series, 364 of polynomials, 155 of segment in given ratio, 82 divisor, greatest common, 4 domain boundary, 401 closed, 402 formula for calculation of volume, 327 multiply connected, 409 of attraction of fixed point, 877 of function, 236 on complex plane, 401 simply connected, 409 double factorial, 10 double integral, 317 change of variables, 321 estimation theorem, 318 geometric applications, 321 geometric meaning, 317 INDEX 1465 double integral (continued) mean value theorem, 318 physical applications, 321 properties, 317 double point, 371 dual linear programming problem, 1017 duality theorem, second, 1018 Duffing equation, 501 Duhamel integral, 516 Duhamel principle, 646 first, 646 second, 648 dynamic programming, 1027 E eccentricity of ellipse, 98 of hyperbola, 102 edge of dihedral angle, 60 of polygon, 51 of polyhedral angle, 61 of polyhedron, 61 of trihedral angle, 60 efficient unbiased estimator, 1089 eigenfunction, 482, 604, 605, 829 integral equation, 835 eigenfunctions Fredholm integral equation, 829 incomplete system, 836 system, 836 eigenvalue, 183, 482, 605, 829 eigenvalue problem, 604 for linear ordinary differential equations, 482 eigenvalues algebraic multiplicity, 209 geometric multiplicity, 209 linear operator, 211, 209 matrix, 183 operator, 209 eigenvector linear operator, 209 matrix, 184 elastic rods, transverse vibration, 1294 electronic mathematical resources, 1451 element admissible, 1009 conjugate, of group, 231 coordinates, 188, 189 inverse, 225 length, 193 neutral, 225 norm, 193 probability, 1040 surface linear, 390 elementary event, 1031 elementary events space, 1031 elementary functions, 19–41 elementary geometry, 43 elementary matrices, 181 elementary symmetric functions, 163 elementary transformation of matrix, 180, 199 of system of linear equations, 199 elements linearly dependent, 188 linearly independent, 188 of linear space, 187 orthogonal, 193 ellipse, 98 canonical equation, 98 center, 99 conjugate diameters, 100 diameter, 100 directrix, 99 eccentricity, 98 focal axis, 99 focal distance, 98 focal parameter, 98 focal property, 99 focus-directrix property, 100 gorge, 145 in polar coordinate system, 101 left directrix, 99 left focus, 99 linear eccentricity, 98 minor axis, 98 numerical eccentricity, 98 optical property, 100 parameter, 98 right directrix, 99 right focus, 99 semimajor axis, 98 semiminor axis, 98 throat, 145 vertex, 99 ellipsoid, 143, 224 imaginary, 224 right, 145 scalene, 144 semiaxis, 143 triaxial, 144 elliptic coordinates, 1202 elliptic equation, 585, 590, 642, 972, 1284 canonical form, 588 first boundary value problem, 633, 643, 644 fundamental solution, 642 mixed boundary value problem, 644 second boundary value problem, 633, 643, 644 third boundary value problem, 643, 644 elliptic equations boundary value problems, 631, 643, 644 linear, 588, 609–611, 631–633, 637, 1284– 1298 nonlinear, 653, 654, 318–1334 elliptic integral, 969 complete, 969 first kind, 970 incomplete, 970 second kind, 970 third kind, 971 elliptic modulus, 971 . 106 of second-order curve, 108 of set, 317 of sphere, 69 partition, 317 principal, of ellipse, 100 principal, of hyperbola, 104 principal, of parabola, 106 diameters conjugate, 151 conjugate, of. 648 dynamic programming, 1027 E eccentricity of ellipse, 98 of hyperbola, 102 edge of dihedral angle, 60 of polygon, 51 of polyhedral angle, 61 of polyhedron, 61 of trihedral angle, 60 efficient unbiased. 1088 standard, 1045 diagonal main, 168 secondary, 168 diagonal matrix, 168 diameter, 151 conjugate to chords, 108 conjugate to family of planes, 151 of circle, 56 of ellipse, 100 of hyperbola, 104 of