Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXdifference, finite, see finite differencesdifferentiablefunction of a complex variable, 825–827function of a real variable, 42differentialdefinition, 43exact and inexact, 155of vector, 338, 344total, 154differential equations, see ordinary differentialequations and partial differential equationsdifferential equations, particularassociated Laguerre, 535, 566, 621–624associated Legendre, 535, 566, 587–593Bernoulli, 477Bessel, 535, 566, 602–607, 614Chebyshev, 535, 566, 595–602Clairaut, 483confluent hypergeometric, 535, 566, 633diffusion, 678, 695–698, 716, 723, 1032Euler, 504Euler–Lagrange, 776Helmholtz, 737–741Hermite, 535, 566, 624–628hypergeometric, 535, 566, 628–632Lagrange, 789Laguerre, 535, 566, 616–621Laplace, 679, 690, 717, 718, 1031Legendre, 534, 535, 566, 577–586Legendre linear, 503–505Poisson, 679, 744–746Schrödinger, 679, 741, 768, 795simple harmonic oscillator, 535, 566Sturm–Liouville, 790wave, 676, 689, 693–695, 714, 737, 790differential operators, see linear differentialoperatordifferentiation, see also derivativeas gradient, 42as rate of change, 41chain rule, 46covariant, 968–971from first principles, 41–44implicit, 47logarithmic, 48notation, 43of Fourier series, 424of integrals, 178of power series, 135partial, see partial differentiationproduct rule, 44–46, 48–50quotient rule, 47theorems, 55–57using complex numbers, 101diffraction, see Fraunhofer diffractiondiffusion equation, 678, 688, 695–698combination of variables, 696–698integral transforms, 747numerical methods, 1032separation of variables, 716simple solution, 696superposition, 723diffusion of solute, 678, 696, 747dihedral group, 1113, 1116dimension of irrep, 1088dimensionality of vector space, 243dipole matrix elements, 208, 1108, 1115dipole moments of molecules, 1077Dirac δ-function, 355, 405, 439–443and convolution, 447and Green’s functions, 511, 512as limit of various distributions, 443as sum of harmonic waves, 442definition, 439Fourier transform of, 443impulses, 441point charges, 441properties, 439reality of, 443relation to Fourier transforms, 442relation to Heaviside (unit step) function, 441three-dimensional, 441, 452Dirac notation, 648direct product, of groups, 1072direct sum ⊕, 1086direction cosines, 221Dirichlet boundary conditions, 702, 852nGreen’s functions, 754, 756–765method of images, 758–765Dirichlet conditions, for Fourier series, 415disc, moment of inertia, 208discontinuous functions and Fourier series,420–422discrete Fourier transforms, 462disjoint events, see mutually exclusive eventsdisplacement kernel, 809distance from aline to a line, 231line to a plane, 232point to a line, 229point to a plane, 230distributive law foraddition of matrix products, 254convolution, 447, 458inner product, 244linear operators, 249multiplicationof a matrix by a scalar, 251of a vector by a complex scalar, 222ofavectorbyascalar,214multiplication by a scalarin a vector space of finite dimensionality,242in a vector space of infinite dimensionality,556scalar or dot product, 220vector or cross product, 222div, divergence of vector fields, 352as integral, 398in curvilinear coordinates, 3671312