Mathematical Tools for Physics - Miami

1 Mathematical Tools for Physicsby James NearingPhysics DepartmentUniversity of 2003, James NearingPermission to copy forindividual or classroomuse is May, 2010 ContentsIntroductioniiiBibliographyv1 Basic Stuff1 TrigonometryParametric DifferentiationGaussian Integralserf and GammaDifferentiatingIntegralsPolar CoordinatesSketching Graphs2 Infinite Series24 The BasicsDeriving Taylor SeriesConvergenceSeries of SeriesPower series, two variablesStirling s ApproximationUseful TricksDiffractionChecking Results3 Complex Algebra52 Complex NumbersSome FunctionsApplications of Euler s FormulaGeometrySeries of cosinesLogarithmsMapping4 Differential Equations67 Linear Constant-CoefficientForced OscillationsSeries SolutionsSome General MethodsTrigonometry via ODE sGreen s FunctionsSeparation of VariablesCircuitsSimultaneous EquationsSimultaneous ODE sLegendre s EquationAsymptotic Behavior5 Fourier Series100 ExamplesComputing Fourier SeriesChoice of BasisMusical NotesPeriodically Forced ODE sReturn to ParsevalGibbs Phenomenon6 Vector Spaces123 The Underlying IdeaAxiomsExamples of Vector SpacesLinear IndependenceNormsScalar ProductBases and Scalar ProductsGram-Schmidt OrthogonalizationCauchy-Schwartz inequalityInfinite Dimensions7 Operators and Matrices143 The Idea of an OperatorDefinition of an OperatorExamples of OperatorsMatrix

2 MultiplicationInversesRotations, 3-dAreas, Volumes, DeterminantsMatrices as OperatorsEigenvalues and EigenvectorsChange of BasisSummation ConventionCan you Diagonalize a Matrix?Eigenvalues and GoogleSpecial Operators8 Multivariable Calculus179 Partial DerivativesChain RuleDifferentialsGeometric InterpretationGradientElectrostaticsPlan e Polar CoordinatesCylindrical, Spherical CoordinatesVectors: Cylindrical, Spherical BasesiGradient in other CoordinatesMaxima, Minima, SaddlesLagrange MultipliersSolid AngleRainbow9 Vector Calculus 1213 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 Equations242 The Heat EquationSeparation of VariablesOscillating TemperaturesSpatial Temperature DistributionsSpecified Heat FlowElectrostaticsCylindrical Coordinates11 Numerical Analysis267 InterpolationSolving equationsDifferentiationIntegrationDiffe rential EquationsFitting of DataEuclidean FitDifferentiating noisy dataPartial Differential Equations12 Tensors294 ExamplesComponentsRelations between TensorsBirefringenceNon-Orthogonal BasesManifolds and FieldsCoordinate BasesBasis Change13 Vector Calculus 2325 IntegralsLine IntegralsGauss s TheoremStokes TheoremReynolds Transport

3 TheoremFields as Vector Spaces14 Complex Variables347 DifferentiationIntegrationPower (Laurent) SeriesCore PropertiesBranch PointsCauchy s Residue TheoremBranch PointsOther IntegralsOther Results15 Fourier Analysis370 Fourier TransformConvolution TheoremTime-Series AnalysisDerivativesGreen s FunctionsSine and Cosine TransformsWiener-Khinchine Theorem16 Calculus of Variations383 ExamplesFunctional DerivativesBrachistochroneFermat s PrincipleElectric FieldsDiscrete VersionClassical MechanicsEndpoint VariationKinksSecond Order17 Densities and Distributions409 DensityFunctionalsGeneralizationDelta-fu nction NotationAlternate ApproachDifferential EquationsUsing Fourier TransformsMore wrote this text for a one semester course at the sophomore-junior level. Our experience withstudents taking our junior Physics courses is that even if they ve had the Mathematical prerequisites,they usually need more experience using the mathematics to handle it efficiently and to possess usableintuition about the processes involved.

4 If you ve seen infinite series in a calculus course, you may haveno idea that they re good for anything. If you ve taken a differential equations course, which of thescores 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 everything be rectangular?How do you learn intuition?When you ve finished a problem and your answer agrees with the back of the book or withyour friends or even a teacher, you re not done. The way to get an intuitive understanding of themathematics and of the Physics is to analyze your solution thoroughly. Does it make sense? Thereare almost always several parameters that enter the problem, so what happens to your solution whenyou push these parameters to their limits? In a mechanics problem, what if one mass is much largerthan another?

5 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 doinga surface integral should 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 your own mistakes before someone else does. Second, you acquire an intuition about howthe equations ought to behave and how the world that they describe ought to behave. Third, It makesall your later efforts easier because you will then have some clue about why the equations work the waythey do. It reifies the it take extra time? Of course. It will however be some of the most valuable extra time youcan it only the students in my classes, or is it a widespread phenomenon that no one is willing tosketch a graph?

6 ( Pulling teeth is the clich e that comes to mind.) Maybe you ve never been taughtthat there are a few basic methods that work, so look at keep referring to isone of those basic Tools that is far more important than you ve ever been told. It is astounding howmany problems become simpler after you ve sketched a graph. Also, until you ve sketched some graphsof functions you really don t know how they I taught this course I didn t do everything that I m presenting here. The two chapters,Numerical Analysis and Tensors, were not in my one semester course, and I didn t cover all of the topicsalong the way. Several more chapters were added after the class was over, so this is now far beyond aone semester text. There is enough here to select from if this is a course text, but if you are readingit on your own then you can move through it as you please, though you will find that the first fivechapters are used more in the later parts than are chapters six and seven.

7 Chapters 8, 9, and 13 form asort of package. I ve tried to use examples that are not all repetitions of the ones in traditional physicstexts but 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 sectionreference to go to that point in the text. To return, there s a Previous View button at the top orbottom of the reader or a keyboard shortcut to do the same thing. [Command on Mac, Alt onWindows, Control on Linux-GNU] The index pages are hyperlinked, and the contents also appear inthe bookmark chose this font for the display versions of the text because it appears better on the screen thandoes the more common 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 thetext.

8 Also Howard Gordon, who used it in his course and provided me with many suggestions forimprovements. Prof. Joseph Tenn of Sonoma State University has given me many very helpful ideas,correcting mistakes, improving notation, and suggesting ways to help the change in notation in this edition: For polar and cylindrical coordinate systems it is common to usetheta for the polar angle in one and phi for the polar angle in the other. I had tried to make them thesame ( ) to avoid confusion, but probably made it less rather than more helpful because it differed fromthe spherical azimuthal coordinate. In this edition all 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, andmany authors will usedSfor an element of area. I have tried to avoid this confusion by sticking tod`anddArespectively (with rare exceptions).

9 In many of the chapters there are exercises that precede the problems. These are supposedto be simpler and mostly 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 electronicversion will remain available Methods for Physics and Engineeringby Riley, Hobson, and Bence. Cambridge Uni-versity PressFor the quantity of well-written material here, it is surprisingly inexpensive in Methods in the Physical Sciencesby Boas. John Wiley PublAbout the right leveland with a very useful selection of topics. If you know everything in here, you ll find all your upper levelcourses much Methods for Physicistsby Arfken and Weber. Academic PressAt a more advancedlevel, but it is sufficiently thorough that will be a valuable reference work Methods in Physicsby Mathews and sophisticated in its approach tothe subject, but it has some beautiful insights.

10 It s considered a standard, though now hard to Methodsby Hassani. SpringerAt the same level as this text with many of the sametopics, but said differently. It is always useful to get a second viewpoint because it s commonly thesecond one that makes sense in whichever order you read s Outlinesby are many good and inexpensive books in this series: forexample, Complex Variables, Advanced Calculus, German Grammar. Amazon lists Complex Analysisby Needham, Oxford University PressThe title tells you the the geometry is paramount, but the traditional material is present too. It s actually fun to read.(Well, I think so anyway.) The Schaum text provides a complementary image of the Analysis for Mathematics and Engineeringby Mathews and Howell. Jones and BartlettPressAnother very good choice for a text on complex variables. Despite the title, mathematiciansshould find nothing wanting Analysisby PublicationsThis publisher has a large selection of moderatelypriced, high quality discursive than most books on numerical analysis, and shows greatinsight into the Differential Operatorsby Lanczos.