ContentsIntroductionChapter I Vector analysisDefinitions, elementary approachAdvanced definitionsScalar or dot productVector or cross productTriple scalar product, triple vector productGradientDivergenceCrulSuccessive aplications of gradVector integrationGausss TheoremStokess TheoremPotential theoryGausss law, Poissons equationHelmhotzs theorem

Chapter II Coordinate systemsCurvilinear coordinatesDifferential vector operationsSpecial coordinate systems- Rectangular cartesian coordinatesCircular cylindrical coordinates (ro,phi,z)Spherical polar coordinates (r,theta,phi)Separation of variables

Chapter III Tensor analysisIntroduction, definitionsContraction, direct productQuotient rulePseudotensors, dual tensorsDyadicsTheory of elasticityLorentz covariance of Maxwells equationsNoncartesian tensors, covariant differentiationTensor differential operations

Chapter IV Determinants, matrices, and group theoryDeterminantsMatricesOrthogonal matricesOblique coordinatesHermitian matrices, unitary matricesDiagonalization of matricesEigenvectors, eigenvaluesIntroduction to group theoryDiscrete groupsContinuous groupsGeneratorsSU(2), SU(3), and nuclear particlesHomogeneus Lorenz group

Chapter V Infinite seriesFundamental conceptsConvergence testsAlternating seriesAlgebra of seriesSeries of functionsTaylors expansionPower seriesElliptic integralsBernoulli numbers, Euler-Maclaurin formulaAsymptotic or semiconvergent seriesInfinite products

Chapter VI Functions of a complex variable IComplex algebraCauchy-Riemann conditionsCauchys Integral theoremCauchys Integral formulaLaurent expansionMappingConformal mapping

Chapter VII Functions of a complex variable II: Calculus of residuesSingularitiesCalculus of residuesDispersion relationsThe method of steepest descents

Chapter VIII Differential equationsPartial differential equations fo theoretical physicsFrist-order differential equationsSeparation of variables- ordinary differential equationsSingular pointsSeries solutions -Frobeniuss MethodA second solutionNonhomogeneus equation- Greens functionNumerical solutions

Chapter IX Strum-Luiville theory-orthogonal functionsSelf-adjoint differential equationsHermitian (Self-adjoint) operatorsGram-Schmidt orthogonalizationCompleteness of eigenfunctions

Chapter X The gamma function (factorial function)Definitions, simple propertiesDigamma and polygamma functionsStirlings seriesThe beta functionThe incomplete gamma functions and related functions

Chapter XI Bessel functionsBessel functions of the first kind, Jv(x)OrthogonalityNeumann functions, Bessel functions of the second kind, Nv(x)Hankel functionsModified Bessel functions, Iv(x) and Kv(x)Asymptotic expansionsSpherical Bessel functions

Chapter XII Legendre functionsGenerating functionRecurrence relations and special propertiesOrthogonalityAlternate definitions of Legendre polynomialsAssociated Legendre functionsSpherical harmonicsAngular momentum ladder operatorsThe addition theorem for spherical harmonicsIntegrals of the product of three spherical harmonicsLegendre functions of the second kind, Qn(x)Vector spherical harmonics

Chapter XIII Special functionsHermite functionsLaguerre functionsChebyshev (Tschevyscheff) polynomialsChebyshev polynomials-numerical applicationsHypergeometric functionsConfluent hypergeometric functions

Chapter XIV Fourier seriesGeneral propertiesAdventages, uses of Fourier seriesApplications of Fourier seriesProperties of Fourier seriesGibbs phenomenonDiscrete orthogonality-discrete Fourier transform

Chapter XV Integral transformsIntegral transformsDevelopment of the Fourier integralFourier transforms-Inversion theoremFourier transform of DerivatesConvolution theoremMomentum representationTransfer functionsElementary Laplace transformsLaplace transform of derivatesOther propertiesConvolution or Faltung theoremInverse Laplace transformation

Chapter XVI Integral equationsIntroductionIntegral transforms, generating functionsNeumann series, separable (degenerate) kernelsHilbert-Schmidt theoryGreens functions-one dimensionGreens functions-two and three dimensions

Chapter XVII Calculus of variationsOne-dependent and one-independent variableApplications of the Euler equationGeneralizations, several dependent variablesSeveral independent variablesMore than one dependent, more than one independent variableLagrangian multiplersVariation subject to constraintsRayleigh-Ritz variational technique

Appendix 1 Real zeros of a functionAppendix 2 Gaussian quadratureGeneral referencesIndex