Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXtranspose, 250matrices, properties ofanti- or skew-symmetric, 270anti-Hermitian, see anti-Hermitian matricesdeterminant, see determinantsdiagonal, 268eigenvalues, see eigenvalueseigenvectors, see eigenvectorsHermitian, see Hermitian matricesnormal, see normal matricesnullity, 293order, 249orthogonal, 270rank, 267square, 249symmetric, 270trace or spur, 258triangular, 269tridiagonal, 998–1000, 1030unitary, see unitary matricesmatrix elements in quantum mechanicsas integrals, 1103dipole, 1108, 1115maxima and minima (local) of a function ofconstrained variables, see Lagrangeundetermined multipliersone real variable, 50–52sufficient conditions, 51several real variables, 162–167sufficient conditions, 164, 167maximum modulus theorem, 881maximum-likelihood, method of, 1255–1271and Bayesian approach, 1264bias, 1260data modelling, 1255estimator, 1256extended, 1270log-likelihood function, 1258parameter estimation, 1255transformation invariance, 1260Maxwell’selectromagnetic equations, 373, 408, 979thermodynamic relations, 176–178Maxwell–Boltzmann statistics, 1138mean µfrom MGF, 1163from PGF, 1158of RVD, 1144of sample, 1223of sample: geometric, harmonic, root meansquare, 1223mean value of a function ofone variable, 72several variables, 199mean value theorem, 56median of RVD, 1145membranedeformed rim, 725–727normal modes, 739, 1112transverse vibrations, 677, 739, 768, 1112method of images, 706, 758–765, 878disc (section of cylinder), 764, 766infinite plate, 759intersecting plates in two dimensions, 761sphere, 762–764, 772metric tensor, 957–960, 963and Christoffel symbols, 966and scale factors, 957, 972covariant derivative of, 982determinant, 957, 964derivative of, 973length element, 957raising or lowering index, 959, 963scalar product, 958volume element, 957, 981MGF, see moment generating functionsMilne’s method, 1022minimum-variance estimator, 1232minor of a matrix element, 259mixed, components of tensor, 957, 962, 969ML estimators, 1256bias, 1260confidence limits, 1262efficiency, 1261transformation invariance, 1260mode of RVD, 1145modulo, mod N, multiplication, 1049modulusof a complex number, 87of a vector, see magnitude of a vectormoleculesbonding in, 1103, 1105–1108dipole moments of, 1077symmetries of, 1077moment generating functions (MGFs),1162–1167and central limit theorem, 1195and PGF, 1163mean and variance, 1163particular distributionsbinomial, 1170exponential, 1191Gaussian, 1163, 1185Poisson, 1177properties, 1163moments (of distributions)central, 1148of RVD, 1147moments (of forces), vector representation of,223moments of inertiaand inertia tensor, 951definition, 198of disc, 208of rectangular lamina, 198of right circular cylinder, 209of sphere, 205perpendicular axes theorem, 209momentum as first-order tensor, 933monomorphism, 10611322