物理学和工程学中的数学方法

Prefacetothesecondedition
Prefacetothefirstedition
1Preliminaryalgebra
1.1Simplefunctionsandequations
Polynomialequations;factorisation;propertiesofroots
1.2Trigonometricidentities
Singleangle;compound-angles;double-andhalf-angleidentities
1.3Coordinategeometry
1.4Partialfractions
Complicationsandspecialcases
1.5Binomialexpansion
1.6Propertiesofbinomialcoefficients
1.7Someparticularmethodsofproof
Proofbyinduction;proofbycontradiction;necessaryandsufficientconditions
1.8Exercises
1.9Hintsandanswers
2Preliminarycalculus
2.1Differentiation
Differentiationfromfirstprinciples:products;thechainrule;quotients;implicitdifferentiation;logarithmicdifferentiation;Leibnitz'theorem;specialpointsofafunction:curvature:theoremsofdifferentiation
2.2Integration
Integrationfromfirstprinciples;theinverseofdifferentiation;byinspection;sinusoidalJhnctions;logarithmicintegration;usingpartialfractions;substitutionmethod;integrationbyparts;reductionformulae;infiniteandimproperintegrals;planepolarcoordinates;integralinequalities;applicationsofintegration
2.3Exercises
2.4Hintsandanswers
3Complexnumbersandhyperbolicfunctions
3.1Theneedforcomplexnumbers
3.2Manipulationofcomplexnumbers
Additionandsubtraction;modulusandargument;multiplication;complexconjugate;division
3.3PolarrepresentationofcomplexnumbersMultiplicationanddivisioninpolarform
3.4deMoivre'stheorem
trigonometricidentities;findingthenthrootsofunity:solvingpolynomialequations
3.5Complexlogarithmsandcomplexpowers
3.6Applicationstodifferentiationandintegration
3.7Hyperbolicfunctions
Definitions;hyperbolic-trigonometricanalogies;identitiesofhyperbolicfunctions:solvinghyperbolicequations;inversesofhyperbolicfunctions;calculusofhyperbolicfunctions
3.8Exercises
3.9Hintsandanswers
4Seriesandlimits
4.1Series
4.2Summationofseries
Arithmeticseries;geometricseries;arithmetico-geometricseries;thedifferencemethod;seriesinvolvingnaturalnumbers;transformationofseries
4.3Convergenceofinfiniteseries
Absoluteandconditionalconvergence;seriescontainingonlyrealpositiveterms;alternatingseriestest
4.4Operationswithseries
4.5Powerseries
Convergenceofpowerseries;operationswithpowerseries
4.6Taylorseries
Taylor'stheorem;approximationerrors;standardMaclaurinseries
4.7Evaluationoflimits
4.8Exercises
4.9Hintsandanswers
5Partialdifferentiation
5.1Definitionofthepartialderivative
5.2Thetotaldifferentialandtotalderivative
5.3Exactandinexactdifferentials
5.4Usefultheoremsofpartialdifferentiation
5.5Thechainrule
5.6Changeofvariables
5.7Taylor'stheoremformany-variablefunctions
5.8Stationaryvaluesofmany-variablefunctions
5.9Stationaryvaluesunderconstraints
5.10Envelopes
5.11Thermodynamicrelations
5.12Differentiationofintegrals
5.13Exercises
5.14Hintsandanswers
6Multipleintegrals
6.1Doubleintegrals
6.2Tripleintegrals
6.3Applicationsofmultipleintegrals
Areasandvolumes;masses,centresofmassandcentroids;Pappus'theorems;momentsofinertia;meanvaluesoffunctions
6.4Changeofvariablesinmultipleintegrals
Changeofvariablesindoubleintegrals;evaluationoftheintegralI=changeofvariablesintripleintegrals;generalpropertiesofJacobians
6.5Exercises
6.6Hintsandanswers
7Vectoralgebra
7.1Scalarsandvectors
7.2Additionandsubtractionofvectors
7.3Multiplicationbyascalar
7.4Basisvectorsandcomponents
7.5Magnitudeofavector
7.6Multiplicationofvectors
Scalarproduct;vectorproduct;scalartripleproduct;vectortripleproduct
7.7Equationsoflines,planesandspheres
7.8Usingvectorstofinddistances
Pointtoline;pointtoplane;linetoline;linetoplane
7.9Reciprocalvectors
7.10Exercises
7.11Hintsandanswers
8Matricesandvectorspaces
8.1Vectorspaces
Basisvectors;innerproduct;someusefulinequalities
8.2Linearoperators
8.3Matrices
8.4Basicmatrixalgebra
Matrixaddition;multiplicationbyascalar;matrixmultiplication
8.5Functionsofmatrices
8,6Thetransposeofamatrix
8.7ThecomplexandHermitianconjugatesofamatrix
8.8Thetraceofamatrix
8.9Thedeterminantofamatrix
Propertiesofdeterminants
8.10Theinverseofamatrix
8.11Therankofamatrix
8.12Specialtypesofsquarematrix
Diagonal;triangular;symmetricandantisymmetric;orthogonal;Hermitianandanti-Hermitian;unitary;normal
8.13Eigenvectorsandeigenvalues
Oranormalmatrix;ofHermitianandanti~Herrnitianmatrices;oraunitarymatrix;orageneralsquarematrix
8.14Determinationofeigenvaluesandeigenvectors
Degenerateeigenvalues
8.15Changeofbasisandsimilaritytransformations
8.16Diagonalisationofmatrices
8.17QuadraticandHermitianforms
Stationarypropertiesoftheeigenvectors;quadraticsurfaces
8.18Simultaneouslinearequations
Range;nullspace;NsimultaneouslinearequationsinNunknowns;singularvaluedecomposition
8.19Exercises
8.20Hintsandanswers
9Normalmodes
9.1Typicaloscillatorysystems
9.2Symmetryandnormalmodes
9.3Rayleigh-Ritzmethod
9.4Exercises
9.5Hintsandanswers
10Vectorcalculus
10.1Differentiationofvectors
Compositevectorexpressions;differentialofavector
10.2Integrationofvectors
10.3Spacecurves
10.4Vectorfunctionsofseveralarguments
10.5Surfaces
10.6Scalarandvectorfields
10.7Vectoroperators
Gradientofascalarfield:divergenceofavectorfield:curlofavectorfield
10.8Vectoroperatorformulae
Vectoroperatorsactingonsumsandproducts;combinationsofgrad,divandcurl
10.9Cylindricalandsphericalpolarcoordinates
10.10Generalcurvilinearcoordinates
10.11Exercises
10.12Hintsandanswers
11Line,surfaceandvolumeintegrals
11.1Lineintegrals
Evaluatinglineintegrals;physicalexamples;lineintegralswithrespecttoascalar
11.2Connectivityofregions
11.3Green'stheoreminaplane
11.4Conservativefieldsandpotentials
11.5Surfaceintegrals
Evaluatingsurfaceintegrals;vectorareasofsurfaces;physicalexamples
11.6Volumeintegrals
Volumesofthree-dimensionalregions
11.7Integralformsforgrad,divandcurl
11.8Divergencetheoremandrelatedtheorems
Green'stheorems;otherrelatedintegraltheorems;physicalapplications
11.9Stokes'theoremandrelatedtheorems
Relatedintegraltheorems:physicalapplications
11.10Exercises
11.11Hintsandanswers
12Fourierseries
12.1TheDirichletconditions
12.2TheFouriercoefficients
12.3Symmetryconsiderations
12.4Discontinuousfunctions
12.5Non-periodicfunctions
12.6Integrationanddifferentiation
12.7ComplexFourierseries
12.8Parseval'stheorem
12.9Exercises
12.10Hintsandanswers
13Integraltransforms
13.1Fouriertransforms
Theuncertaintyprinciple;Fraunhoferdiffraction:theDirac&-function:relationofthe6-functiontoFouriertransforms;propertiesofFouriertransJorms;oddandevenfunctions;convolutionanddeconvolution;correlationfunctionsandenergyspectra;Parseval'stheorem;Fouriertransformsinhigherdimensions
13.2Laplacetransforms
Laplacetransformsofderivativesandintegrals;otherpropertiesofLaplacetransforms
13.3Concludingremarks
13.4Exercises
13.5Hintsandanswers
14First-orderordinarydifferentialequations
14.1Generalformofsolution
14.2First-degreefirst-orderequations
Separable-variableequations;exactequations;inexactequations,integratingfactors;linearequations;homogeneousequations;isobaricequations:Bernoulli'sequation;miscellaneousequations
14.3Higher-degreefirst-orderequations
Equationssolubleforp;forx;fory;Clairaut'sequation
14.4Exercises
14.5Hintsandanswers
15Higher-orderordinarydifferentialequations
15.1Linearequationswithconstantcoefficients
Findingthecomplementaryfunctionyc(x):findingtheparticularintegralyp(x);constructingthegeneralsolutionye(x)+yp(x):linearrecurrencerelations:Laplacetransformmethod
15.2Linearequationswithvariablecoefficients
TheLegendreandEulerlinearequations;exactequations;partiallyknowncomplementaryfunction;variationofparameters;Green'sfunctions;canonicalformforsecond-orderequations
15.3Generalordinarydifferentialequations
Dependentvariableabsent;independentvariableabsent;non-linearexactequations;isobaricorhomogeneousequations;equationshomogeneousinxoryalone;equationshavingy=Aexasasolution
15.4Exercises
15.5Hintsandanswers
16Seriessolutionsofordinarydifferentialequations
16.1Second-orderlinearordinarydifferentialequations
Ordinaryandsingularpoints
16.2Seriessolutionsaboutanordinarypoint
16.3Seriessolutionsaboutaregularsingularpoint
Distinctrootsnotdifferingbyaninteger;repeatedrootoftheindicialequation;distinctrootsdifferingbyaninteger
16.4Obtainingasecondsolution
TheWronskianmethod;thederivativemethod;seriesformofthesecondsolution
16.5Polynomialsolutions
16.6Legendre'sequation
Generalsolutionforinteger1;propertiesofLegendrepolynomials
16.7Bessersequation
Generalsolutionfornon-integerv;generalsolutionforintegerv;propertiesofBesselfunctions
16.8Generalremarks
16.9Exercises
16.10Hintsandanswers
17Eigenfunctionmethodsfordifferentialequations
17.1Setsoffunctions
Someusefulinequalities
17.2AdjointandHermitianoperators
17.3ThepropertiesofHermitianoperators
Realityoftheeigenvalues;orthogonalityoftheeigenfunctions;constructionofrealeigenfunctions
17.4Sturm-Liouvilleequations
Validboundaryconditions;puttinganequationintoSturm-Liouvilleform
17.5ExamplesofSturm-Liouvilleequations
Legendre'sequation;theassociatedLegendreequation;Bessel'sequation;thesimpleharmonicequation;Hermite'sequation;Laguerre'sequation;Chebyshev'sequation
17.6Superpositionofeigenfunctions:Green'sfunctions
17.7Ausefulgeneralisation
17.8Exercises
17.9Hintsandanswers
18Partialdifferentialequations:generalandparticularsolutions
18.1Importantpartialdifferentialequations
Thewaveequation;thediffusionequation;Laplace'sequation;Poisson'sequation;SchrOdinger'sequation
18.2Generalformofsolution
18.3Generalandparticularsolutions
First-orderequations;inhomogeneousequationsandproblems;second-orderequations
18.4Thewaveequation
18.5Thediffusionequation
18.6Characteristicsandtheexistenceofsolutions
First-orderequations;second-orderequations
18.7Uniquenessofsolutions
18.8Exercises
18.9Hintsandanswers
19Partialdifferentialequations:separationofvariablesandothermethods
19.1Separationofvariables:thegeneralmethod
19.2Superpositionofseparatedsolutions
19.3Separationofvariablesinpolarcoordinates
Laplace'sequationinpolarcoordinates:sphericalharmonics:otherequationsinpolarcoordinates;solutionbyexpansion;separationofvariablesforinhomogeneousequations
19.4Integraltransformmethods
19.5Inhomogeneousproblems-Green'sfunctions
SimilaritiestoGreen'sfunctionsforordinarydifferentialequations:generalboundary-valueproblems:Dirichletproblems;Neumannproblems
19.6Exercises
19.7Hintsandanswers
20Complexvariables
20.1Functionsofacomplexvariable
20.2TheCauchy-Riemannrelations
20.3Powerseriesinacomplexvariable
20.4Someelementaryfunctions
20.5Multivaluedfunctionsandbranchcuts
20.6Singularitiesandzeroesofcomplexfunctions
20.7Complexpotentials
20.8Conformaltransformations
20.9Applicationsofconformaltransformations
20.10Complexintegrals
20.11Cauchy'stheorem
20.12Cauchy'sintegralformula
20.13TaylorandLaurentseries
20.14Residuetheorem
20.15Locationofzeroes
20.16Integralsofsinusoidalfunctions
20.17Someinfiniteintegrals
20.18Integralsofmultivaluedfunctions
20.19Summationofseries
20.20InverseLaplacetransform
20.21Exercises
20.22Hintsandanswers
21Tensors
21.1Somenotation
21.2Changeofbasis
21.3Cartesiantensors
21.4First-andzero-orderCartesiantensors
21.5Second-andhigher-orderCartesiantensors
21.6Thealgebraoftensors
21.7Thequotientlaw
21.8Thetensorsand
21.9Isotropictensors
21.10Improperrotationsandpseudotensors
21.11Dualtensors
21.t2Physicalapplicationsoftensors
21.13Integraltheoremsfortensors
21.14Non-Cartesiancoordinates
21.15Themetrictensor
21.16Generalcoordinatetransformationsandtensors
21.17Relativetensors
21.18DerivativesofbasisvectorsandChristoffelsymbols
21.19Covariantdifferentiation
21.20Vectoroperatorsintensorform
21.21Absolutederivativesalongcurves
21.22Geodesics
21.23Exercises
21.24Hintsandanswers
22Calculusofvariations
22.1TheEuler-Lagrangeequation
22.2Specialcases
Fdoesnotcontainyexplicitly;Fdoesnotcontainxexplicitly
22.3Someextensions
Severaldependentvariables;severalindependentvariables;higher-orderderivatives:variableend-points
22.4Constrainedvariation
22.5Physicalvariationalprinciples
Fermat'sprincipleinoptics;Hamilton'sprincipleinmechanics
22.6Generaleigenvalueproblems
22.7Estimationofeigenvaluesandeigenfunctions
22.8Adjustmentofparameters
22.9Exercises
22.10Hintsandanswers
23Integralequations
23.1Obtaininganintegralequationfromadifferentialequation
23.2Typesofintegralequation
23.3Operatornotationandtheexistenceofsolutions
23.4Closed-formsolutions
Separablekernels;integraltransformmethods;differentiation
23.5Neumannseries
23.6Fredholmtheory
23.7Schmidt-Hilberttheory
23.8Exercises
23.9Hintsandanswers
24Grouptheory
24.1Groups
Definitionofagroup;examplesofgroups
24.2Finitegroups
24.3Non-Abeliangroups
24.4Permutationgroups
24.5Mappingsbetweengroups
24.6Subgroups
24.7Subdividingagroup
Equivalencerelationsandclasses;congruenceandcosets;conjugatesandclasses
24.8Exercises
24.9Hintsandanswers
25Representationtheory
25.1Dipolemomentsofmolecules
25.2Choosinganappropriateformalism
25.3Equivalentrepresentations
25.4Reducibilityofarepresentation
25.5Theorthogonalitytheoremforirreduciblerepresentations
25.6Characters
Orthogonalitypropertyofcharacters
25.7Countingirrepsusingcharacters
Summationrulesforirreps
25.8Constructionofacharactertable
25.9Groupnomenclature
25.10Productrepresentations
25.11Physicalapplicationsofgrouptheory
Bondinginmolecules:matrixelementsinquantummechanics:degeneracyofnormalmodes:breakingofdegeneracies
25.12Exercises
25.13Hintsandanswers
26Probability
26.1Venndiagrams
26.2Probability
Axiomsandtheorems;conditionalprobability;Bayes'theorem
26.3Permutationsandcombinations
26.4Randomvariablesanddistributions
Discreterandomvariables;continuousrandomvariables
26.5Propertiesofdistributions
Mean:modeandmedian:varianceandstandarddeviation:moments:
centralmoments
26.6Functionsofrandomvariables
2617Generatingfunctions
Probabilitygeneratingfunctions;momentgeneratingfunctions;characteristicfunctions;cumulantgeneratingfunctions
26.8Importantdiscretedistributions
Binomial;geometric;negativebinomial;hypergeometric;Poisson
26.9Importantcontinuousdistributions
Gaussian:log-normahexponential;gamma;chi-squared;Cauchy;BreitWigner:uniform
26.10Thecentrallimittheorem
26.11Jointdistributions
Discretebivariate;continuousbivariate;marginalandconditionaldistributions
26.12Propertiesofjointdistributions
Means;variances;covarianceandcorrelation
26.13Generatingfunctionsforjointdistributions
26.14Transformationofvariablesinjointdistributions
26.15Importantjointdistributions
MultinominahmultivariateGaussian
26.16Exercises
26.17Hintsandanswers
27Statistics
27.1Experiments,samplesandpopulations
27.2Samplestatistics
Averages;varianceandstandarddeviation;moments;covarianceandcorrelation
27.3Estimatorsandsamplingdistributions
Consistency,biasandefficiency;Fisher'sinequality:standarderrors;confidencelimits
27.4Somebasicestimators
Mean;variance:standarddeviation;moments;covarianceandcorrelation
27.5Maximum-likelihoodmethod
MLestimator;trans]ormationinvarianceandbias;efficiency;errorsandconfidencelimits;Bayesianinterpretation;large-Nbehaviour;extendedMLmethod
27.6Themethodofleastsquares
Linearleastsquares;non-linearleastsquares
27.7Hypothesistesting
Simpleandcompositehypotheses;statisticaltests;Neyman-Pearson;generalisedlikelihood-ratio:Student'st:Fisher'sF:goodnessoffit
27.8Exercises
27.9Hintsandanswers
28Numericalmethods
28.1Algebraicandtranscendentalequations
Rearrangementoftheequation;linearinterpolation;binarychopping;Newton-Raphsonmethod
28.2Convergenceofiterationschemes
28.3Simultaneouslinearequations
Gaussianelimination;Gauss-Seideliteration;tridiagonalmatrices
28.4Numericalintegration
Trapeziumrule;Simpson'srule;Gaussianintegration;MonteCarlomethods
28.5Finitedifferences
28.6Differentialequations
Differenceequations;Taylorseriessolutions;predictionandcorrection;Runge-Kuttamethods;isoclines
28.7Higher-orderequations
28.8Partialdifferentialequations
28.9Exercises
28.10Hintsandanswers
AppendixGamma,betaanderrorfunctions
A1.1Thegammafunction
Al.2Thebetafunction
Al.3Theerrorfunction
Index