Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXnull operation, as identity element of group,1044nullity, of a matrix, 293numerical methods for algebraic equations,985–992binary chopping, 990convergence of iteration schemes, 992–994linear interpolation, 988Newton–Raphson, 990–992rearrangement methods, 987numerical methods for integration, 1000–1009Gaussian integration, 1005–1009mid-point rule, 1034Monte Carlo, 1009nomenclature, 1001Simpson’s rule, 1004trapezium rule, 1002–1004numerical methods for ordinary differentialequations, 1020–1030accuracy and convergence, 1021Adams method, 1024difference schemes, 1021–1023Euler method, 1021first-order equations, 1021–1028higher-order equations, 1028–1030isoclines, 1028Milne’s method, 1022prediction and correction, 1024–1026reduction to matrix form, 1030Runge–Kutta methods, 1026–1028Taylor series methods, 1023numerical methods for partial differentialequations, 1030–1032diffusion equation, 1032Laplace’s equation, 1031minimising error, 1032numerical methods for simultaneous linearequations, 994–1000Gauss–Seidel iteration, 996–998Gaussian elimination with interchange, 995matrix form, 994–1000tridiagonal matrices, 998–1000O(x), order of, 132observables in quantum mechanics, 277, 560odd functions, see antisymmetric functionsODE, see ordinary differential equations (ODEs)operatorsHermitian, see Hermitian operatorslinear, see linear operators and lineardifferential operator and linear integraloperatoroperators (quantum)angular momentum, 656–663annihilation and creation, 667coordinate-free, 648–671eigenvalues and eigenstates, 649physical examplesangular momentum, 658Hamiltonian, 657order ofapproximation in Taylor series, 137nconvergence of iteration schemes, 993group, 1043group element, 1047ODE, 468permutation, 1058recurrence relations (series), 497subgroup, 1061and Lagrange’s theorem, 1065tensor, 930ordinary differential equations (ODE), see alsodifferential equations, particularboundary conditions, 468, 470, 501complementary function, 491degree, 468dimensionally homogeneous, 475exact, 472, 505first-order, 468–484first-order higher-degree, 480–484soluble for p, 480soluble for x, 481soluble for y, 482general form of solution, 468–470higher-order, 490–523homogeneous, 490inexact, 473isobaric, 476, 521linear, 474, 490–517non-linear, 518–523exact, 519isobaric (homogeneous), 521x absent, 518y absent, 518order, 468ordinary point, see ordinary points of ODEparticular integral (solution), 469, 492, 494singular point, see singular points of ODEsingular solution, 469, 481, 482, 484ordinary differential equations, methods forcanonical form for second-order equations,516eigenfunctions, 554–573equations containing linear forms, 478–480equations with constant coefficients, 492–503Green’s functions, 511–516integrating factors, 473–475Laplace transforms, 501–503numerical, 1020–1030partially known CF, 506separable variables, 471series solutions, 531–550, 604undetermined coefficients, 494variation of parameters, 508–510ordinary points of ODE, 533, 535–538indicial equation, 543orthogonal lines, condition for, 12orthogonal matrices, 270, 929, 930general properties, see unitary matricesorthogonal systems of coordinates, 3641324