Mathematical Tools for Physics

1 Mathematical Tools for Physicsby James NearingPhysics DepartmentUniversity of 2003, James NearingPermission to copy forindividual or classroomuse is May, 2010 ContentsIntroductionivBibliographyvi1 Basic Stuff1 TrigonometryParametric DifferentiationGaussian Integralserf and GammaDifferentiatingIntegralsPolar CoordinatesSketching Graphs2 Infinite Series31 The BasicsDeriving Taylor SeriesConvergenceSeries of SeriesPower series, two variablesStirling s ApproximationUseful TricksDiffractionChecking Results3 Complex Algebra69 Complex NumbersSome FunctionsApplications of Euler s FormulaGeometrySeries of cosinesLogarithmsMapping4 Differential Equations88 Linear Constant-CoefficientForced OscillationsSeries SolutionsSome General MethodsTrigonometry via ODE sGreen s FunctionsSeparation of VariablesCircuitsSimultaneous EquationsSimultaneous ODE sLegendre s EquationAsymptotic Behavior5 Fourier Series131 ExamplesComputing Fourier SeriesChoice of BasisMusical NotesPeriodically Forced ODE sReturn to ParsevalGibbs Phenomenon6 Vector Spaces162 The Underlying IdeaAxiomsExamples of Vector SpacesLinear IndependenceNormsScalar ProductBases and Scalar ProductsGram-Schmidt OrthogonalizationiCauchy-Schwartz inequalityInfinite Dimensions7 Operators and

2 Matrices188 The Idea of an OperatorDefinition of an OperatorExamples of OperatorsMatrix MultiplicationInversesRotations, 3-dAreas, Volumes, DeterminantsMatrices as OperatorsEigenvalues and EigenvectorsChange of BasisSummation ConventionCan you Diagonalize a Matrix?Eigenvalues and GoogleSpecial Operators8 Multivariable Calculus235 Partial DerivativesChain RuleDifferentialsGeometric InterpretationGradientElectrostaticsPlan e Polar CoordinatesCylindrical, Spherical CoordinatesVectors: Cylindrical, Spherical BasesGradient in other CoordinatesMaxima, Minima, SaddlesLagrange MultipliersSolid AngleRainbow9 Vector Calculus 1281 Fluid FlowVector DerivativesComputing the divergenceIntegral Representation of CurlThe GradientShorter Cut for div and curlIdentities for Vector OperatorsApplications to GravityGravitational PotentialIndex NotationMore Complicated Potentials10 Partial Differential Equations320 The Heat EquationSeparation of VariablesOscillating TemperaturesSpatial Temperature DistributionsSpecified Heat FlowElectrostaticsCylindrical Coordinates11 Numerical Analysis353 InterpolationSolving equationsDifferentiationIntegrationDiffe rential EquationsFitting of DataEuclidean FitDifferentiating noisy dataPartial Differential Equations12 Tensors391 ExamplesComponentsiiRelations between TensorsBirefringenceNon-Orthogonal BasesManifolds and FieldsCoordinate BasesBasis

3 Change13 Vector Calculus 2430 IntegralsLine IntegralsGauss s TheoremStokes TheoremReynolds Transport TheoremFields as Vector Spaces14 Complex Variables460 DifferentiationIntegrationPower (Laurent) SeriesCore PropertiesBranch PointsCauchy s Residue TheoremBranch PointsOther IntegralsOther Results15 Fourier Analysis492 Fourier TransformConvolution TheoremTime-Series AnalysisDerivativesGreen s FunctionsSine and Cosine TransformsWiener-Khinchine Theorem16 Calculus of Variations510 ExamplesFunctional DerivativesBrachistochroneFermat s PrincipleElectric FieldsDiscrete VersionClassical MechanicsEndpoint VariationKinksSecond Order17 Densities and Distributions545 DensityFunctionalsGeneralizationDelta-fu nction NotationAlternate ApproachDifferential EquationsUsing Fourier TransformsMore wrote this text for a one semester course at the sophomore-junior level.

4 Our experience with students takingour junior Physics courses is that even if they ve had the Mathematical prerequisites, they usually need more experienceusing the mathematics to handle it efficiently and to possess usable intuition about the processes involved. If you ve seeninfinite series in a calculus course, you may have no idea that they re good for anything. If you ve taken a differentialequations course, which of the scores of techniques that you ve seen are really used a lot?The world is (at least) three dimensional so you clearly need to understand multiple integrals, but will everythingbe rectangular?How do you learn intuition?When you ve finished a problem and your answer agrees with the back of the book or with your friends or even ateacher, you re not done. The way to get an intuitive understanding of the mathematics and of the Physics is to analyzeyour solution thoroughly.

5 Does it make sense? There are almost always several parameters that enter the problem, sowhat happens to your solution when you push these parameters to their limits? In a mechanics problem, what if onemass is much larger than another? Does your solution do the right thing? In electromagnetism, if you make a couple ofparameters equal to each other does it reduce everything to a simple, special case? When you re doing a surface integralshould the answer be positive or negative and does your answer agree?When you address these questions to every problem you ever solve, you do several things. First, you ll find yourown mistakes before someone else does. Second, you acquire an intuition about how the equations ought to behave andhow the world that they describe ought to behave.

6 Third, It makes all your later efforts easier because you will then havesome clue about why the equations work the way they do. It reifies the it take extra time? Of course. It will however be some of the most valuable extra time you can it only the students in my classes, or is it a widespread phenomenon that no one is willing to sketch a graph?( Pulling teeth is the clich e that comes to mind.) Maybe you ve never been taught that there are a few basic methodsthat work, so look at keep referring to is one of those basic Tools that is far more importantthan you ve ever been told. It is astounding how many problems become simpler after you ve sketched a graph. Also,until you ve sketched some graphs of functions you really don t know how they I taught this course I didn t do everything that I m presenting here.

7 The two chapters, Numerical Analysisand Tensors, were not in my one semester course, and I didn t cover all of the topics along the way. Several more chapterswere added after the class was over, so this is now far beyond a one semester text. There is enough here to select fromif this is a course text, but if you are reading it on your own then you can move through it as you please, though youwill find that the first five chapters are used more in the later parts than are chapters six and seven. Chapters 8, 9, andiv13 form a sort of package. I ve tried to use examples that are not all repetitions of the ones in traditional Physics textsbut that do provide practice in the same Tools that you need in that pdf file that I ve placed online is hyperlinked, so that you can click on an equation or section reference to goto that point in the text.

8 To return, there s a Previous View button at the top or bottom of the reader or a keyboardshortcut to do the same thing. [Command on Mac, Alt on Windows, Control on Linux-GNU] The index pagesare hyperlinked, and the contents also appear in the bookmark chose this font for the display versions of the text because it appears better on the screen than does the morecommon Times font. The choice of available mathematics fonts is more d like to thank the students who found some, but probably not all, of the mistakes in the text. Also HowardGordon, who used it in his course and provided me with many suggestions for improvements. Prof. Joseph Tenn ofSonoma State University has given me many very helpful ideas, correcting mistakes, improving notation, and suggestingways to help the change in notation in this edition: For polar and cylindrical coordinate systems it is common to use theta for the polarangle in one and phi for the polar angle in the other.

9 I had tried to make them the same ( ) to avoid confusion, butprobably made it less rather than more helpful because it differed from the spherical azimuthal coordinate. In this editionall three systems (plane polar, cylindrical, spherical) use phi as = tan 1(y/x). In line integrals it is common to usedsfor an element of length, and many authors will usedSfor an element of area. I have tried to avoid this confusionby sticking tod`anddArespectively (with rare exceptions).In many of the chapters there are exercises that precede the problems. These are supposed to be simpler andmostly designed to establish some of the definitions that appeared in the text is now available in print from Dover Publishers. They have agreed that the electronic version will remainavailable methods for Physics and Engineeringby Riley, Hobson, and Bence.

10 Cambridge University PressForthe quantity of well-written material here, it is surprisingly inexpensive in methods in the Physical Sciencesby Boas. John Wiley PublAbout the right level and with a veryuseful selection of topics. If you know everything in here, you ll find all your upper level courses much methods for Physicistsby Arfken and Weber. Academic PressAt a more advanced level, but it issufficiently thorough that will be a valuable reference work methods in Physicsby Mathews and sophisticated in its approach to the subject, butit has some beautiful insights. It s considered a standard, though now hard to Methodsby Hassani. SpringerAt the same level as this text with many of the same topics, but saiddifferently. It is always useful to get a second viewpoint because it s commonly the second one that makes sense inwhichever order you read s Outlinesby are many good and inexpensive books in this series: for example, ComplexVariables, Advanced Calculus, German Grammar.