Mathematical Tools for Physics

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Mathematical Tools for Physicsby James NearingPhysics DepartmentUniversity of 2003, James NearingPermission to copy forindividual or classroomuse is May, 2010ContentsIntroductioniiiBibliographyv 1Basic Stuff1TrigonometryParametric DifferentiationGaussian Integralserf and GammaDifferentiatingIntegralsPolar CoordinatesSketching Graphs2Infinite Series24The BasicsDeriving Taylor SeriesConvergenceSeries of SeriesPower series, two variablesStirling s ApproximationUseful TricksDiffractionChecking Results3Complex Algebra52Complex NumbersSome FunctionsApplications of Euler s FormulaGeometrySeries of cosinesLogarithmsMapping4Differential Equations67Linear Constant-CoefficientForced OscillationsSeries SolutionsSome General MethodsTrigonometry via ODE sGreen s FunctionsSeparation of VariablesCircuitsSimultaneous EquationsSimultaneous ODE sLegendre s EquationAsymptotic Behavior5Fourier Series100ExamplesComputing Fourier SeriesChoice of BasisMusical NotesPeriodically Forced ODE sReturn to ParsevalGibbs Phenomenon6Vector Spaces123The Underlying IdeaAxiomsExamples of Vector SpacesLinear IndependenceNormsScalar ProductBases and Scalar ProductsGram-Schmidt OrthogonalizationCauchy-Schwartz inequalityInfinite Dim

Parametric Di erentiation Gaussian Integrals erf and Gamma Di erentiating Integrals Polar Coordinates Sketching Graphs 2 In nite Series 24 The Basics Deriving Taylor Series Convergence Series of Series Power series, two variables Stirling’s Approximation Useful Tricks Di raction Checking Results 3 Complex Algebra 52 Complex Numbers Some Functions

  Di erentiation, Erentiation