Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXin spherical polars, 362Stoke’s theorem, 406–409tensor form, 974current-carrying wire, magnetic potential, 729curvature, 52–55circle of, 53of a function, 52of space curves, 342radius of, 53curves, see plane curves and space curvescurvilinear coordinates, 364–369basis vectors, 364length and volume elements, 365scale factors, 364surfaces and curves, 364tensors, 955–977vector operators, 367–369cut plane, 865cycle notation for permutations, 1057cyclic groups, 1061, 1098cyclic relation for partial derivatives, 157cycloid, 370, 785cylinders, conducting, 874, 876cylindrical polar coordinates, 357–361area element, 360basis vectors, 358Laplace equation, 728–731length element, 360vector operators, 357–361volume element, 360δ-function (Dirac), see Dirac δ-functionδ ij , δ j i , Kronecker delta, tensor, see Kroneckerdelta, δ ij , δ j i ,tensorD’Alembert’s ratio test, 126, 832in convergence of power series, 132D’Alembert’s solution to wave equation, 694damped harmonic oscillators, 239and Parseval’s theorem, 451data modelling, maximum-likelihood, 1255de Broglie relation, 436, 709, 768de Moivre’s theorem, 95, 861applications, 95–99finding the nth roots of unity, 97solving polynomial equations, 98trigonometric identities, 95–97deconvolution, 449defective matrices, 278, 311degeneracybreaking of, 1111–1113of normal modes, 1110degenerate (separable) kernel, 807degenerate eigenvalues, 275, 282degreeof ODE, 468of polynomial equation, 2del ∇, see gradient operator (grad)del squared ∇ 2 (Laplacian), 352, 676as integral, 400in curvilinear coordinates, 368in cylindrical polar coordinates, 360in polar coordinates, 725in spherical polar coordinates, 362, 741tensor form, 973delta function (Dirac), see Dirac δ-functiondependent random variables, 1196–1205derivative, see also differentiationabsolute, 975–977covariant, 968Fourier transform of, 444Laplace transform of, 455normal, 350of basis vectors, 336of composite vector expressions, 337of function of a complex variable, 825of function of a function, 46of hyperbolic functions, 106–109of products, 44–46, 48–50of quotients, 47of simple functions, 44of vectors, 334ordinary, first, second and nth, 42partial, see partial differentiationtotal, 154derivative method for second series solution ofODE, 545–548determinant formand ɛ ijk , 942for curl, 353determinants, 259–263adding rows or columns, 262and singular matrices, 263as product of eigenvalues, 287evaluationusing ɛ ijk , 942using Laplace expansion, 259identical rows or columns, 262in terms of cofactors, 259interchanging two rows or two columns, 262Jacobian representation, 201, 205, 207notation, 259of Hermitian conjugate matrices, 262of order three, in components, 260of transpose matrices, 261product rule, 262properties, 261–263, 978relationship with rank, 267removing factors, 262secular, 280diagonal matrices, 268diagonalisation of matrices, 285–288normal matrices, 286properties of eigenvalues, 287simultaneous, 331diamond, unit cell, 234die throwing, see probabilitydifference method for summation of series, 119difference schemes for differential equations,1020–1023, 1030–10321311