VectorIdentities A=A xˆ + A yˆ + A zˆ, A2 = A2+ A2 + A2, A·B= A B + A B + A B x y z x y z x x y y z z (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)A A (cid:2) (cid:2)A A (cid:2) (cid:2)A A (cid:2) A×B=(cid:2) y z(cid:2)xˆ −(cid:2) x z(cid:2)yˆ +(cid:2) x y(cid:2)zˆ (cid:2)B B (cid:2) (cid:2)B B (cid:2) (cid:2)B B (cid:2) y z x z x y (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)Ax Ay Az(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)A A (cid:2) (cid:2)A A (cid:2) (cid:2)A A (cid:2) A·(B×C)=(cid:2)B B B (cid:2)=C (cid:2) y z(cid:2)−C (cid:2) x z(cid:2)+C (cid:2) x y(cid:2) (cid:2) x y z(cid:2) x(cid:2)B B (cid:2) y(cid:2)B B (cid:2) z(cid:2)B B (cid:2) (cid:2) (cid:2) y z x z x y C C C x y z (cid:3) A×(B×C)=BA·C−CA·B, ε ε =δ δ −δ δ ijk pqk ip jq iq jp k VectorCalculus rdV dV df F=−∇V(r)=− =−rˆ , ∇·(rf(r))=3f(r)+r , r dr dr dr ∇·(rrn−1)=(n+2)rn−1 ∇(A·B)=(A·∇)B+(B·∇)A+A×(∇×B)+B×(∇×A) ∇·(SA)=∇S·A+S∇·A, ∇·(A×B)=B·(∇×A)−A·(∇×B) ∇·(∇×A)=0, ∇×(SA)=∇S×A+S∇×A, ∇×(rf(r))=0, ∇×r=0 ∇×(A×B)=A∇·B−B∇·A+(B·∇)A−(A·∇)B, ∇×(∇×A)=∇(∇·A)−∇2A (cid:4) (cid:4) (cid:4) (cid:5) ∇·Bd3r= B·da, (Gauss), (∇×A)·da= A·dl, (Stokes) V S(cid:4) S (cid:4) (φ∇2ψ −ψ∇2φ)d3r = (φ∇ψ −ψ∇φ)·da, (Green) V S 1 1 (cid:3) δ(x−x) ∇2 =−4πδ(r), δ(ax)= δ(x), δ(f(x))= i , r |a| |f(cid:8)(x)| i,f(xi)=0,f(cid:8)(xi)=(cid:9) 0 i (cid:4) (cid:4) 1 ∞ d3k δ(t−x)= eiω(t−x)dω, δ(r)= e−ik·r, 2π (2π)3 −∞ (cid:3)∞ δ(x−t)= ϕ∗(x)ϕ (t) n n n=0 CurvedOrthogonalCoordinates CylinderCoordinates q1 =ρ, q2 =ϕ, q3 =z; h1 =hρ =1, h2 =hϕ =ρ, h3 =hz=1, r=xˆρcosϕ+yˆρsinϕ+zzˆ SphericalPolarCoordinates q1 =r, q2 =θ, q3 =ϕ;h1 =hr =1, h2 =hθ =r, h3 =hϕ =rsinθ, r=xˆrsinθcosϕ+yˆrsinθsinϕ+zˆrcosθ (cid:2) (cid:2) (cid:2) (cid:2) (cid:3) (cid:3) (cid:3) (cid:2)qˆ1 qˆ2 qˆ3(cid:2) (cid:2) (cid:2) dr= h dq qˆ , A= A qˆ , A·B= A B ,A×B=(cid:2)A1 A2 A3(cid:2) i i i i i i i (cid:2) (cid:2) i i i (cid:2)B1 B2 B3(cid:2) (cid:4) (cid:4) (cid:4) (cid:4) (cid:3) f d3r = f(q ,q ,q )h h h dq dq dq F·dr= Fh dq 1 2 3 1 2 3 1 2 3 i i i (cid:4)V (cid:4) (cid:4) L (cid:4) i B·da= B h h dq dq + B h h dq dq + B h h dq dq , 1 2 3 2 3 2 1 3 1 3 3 1 2 1 2 S (cid:3) 1 ∂V ∇V = qˆ , ih ∂q i i i (cid:6) (cid:7) 1 ∂ ∂ ∂ ∇·F= (F h h )+ (F h h )+ (F h h ) h h h ∂q 1 2 3 ∂q 2 1 3 ∂q 3 1 2 1 2 3 1 2 3 (cid:6) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)(cid:7) 1 ∂ h h ∂V ∂ h h ∂V ∂ h h ∂V ∇2V = 2 3 + 1 3 + 2 1 h h h ∂q h ∂q ∂q h ∂q ∂q h ∂q 1 2 3 1 1 1 2 2 2 3 3 3 (cid:2) (cid:2) (cid:2)h qˆ h qˆ h qˆ (cid:2) (cid:2) 1 1 2 2 3 3(cid:2) ∇×F= 1 (cid:2)(cid:2) ∂ ∂ ∂ (cid:2)(cid:2) h h h (cid:2) ∂q1 ∂q2 ∂q3 (cid:2) 1 2 3 (cid:2) (cid:2) h F h F h F 1 1 2 2 3 3 MathematicalConstants e =2.718281828, π =3.14159265, ln10=2.302585093, 1rad=57.29577951◦, 1◦ =0.0174532925 rad, (cid:6) (cid:7) 1 1 1 γ = lim 1+ + +···+ −ln(n+1) =0.577215661901532 n→∞ 2 3 n (Euler-Mascheroninumber) 1 1 1 1 B =− , B = , B = B =− , B = ,... (Bernoullinumbers) 1 2 4 8 6 2 6 30 42 Essential Mathematical Methods for Physicists Essential Mathematical Methods for Physicists Hans J. Weber UniversityofVirginia Charlottesville,VA George B. Arfken MiamiUniversity Oxford,Ohio Amsterdam Boston London NewYork Oxford Paris SanDiego SanFrancisco Singapore Sydney Tokyo SponsoringEditor BarbaraHolland ProductionEditor AngelaDooley EditorialAssistant KarenFrost MarketingManager MarianneRutter CoverDesign RichardHannus PrinterandBinder Quebecor Thisbookisprintedonacid-freepaper.(cid:1)∞ Copyright(cid:3)c 2003,2001,1995,1985,1970,1966byHarcourt/AcademicPress Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyany means,electronicormechanical,includingphotocopy,recording,oranyinformation storageandretrievalsystem,withoutpermissioninwritingfromthepublisher. Requestsforpermissiontomakecopiesofanypartoftheworkshouldbemailedto: PermissionsDepartment,Harcourt,Inc.,6277SeaHarborDrive,Orlando, Florida32887-6777. AcademicPress AHarcourtScienceandTechnologyCompany 525BStreet,Suite1900,SanDiego,CA92101-4495,USA http://www.academicpress.com AcademicPress HarcourtPlace,32JamestownRoad,LondonNW17BY,UK Harcourt/AcademicPress 200WheelerRoad,Burlington,MA01803 http://www.harcourt-ap.com InternationalStandardBookNumber:0-12-059877-9 PRINTEDINTHEUNITEDSTATESOFAMERICA 03 04 05 06 07 Q 9 8 7 6 5 4 3 2 1 Contents Preface xix 1 VECTORANALYSIS 1 1.1 ElementaryApproach 1 VectorsandVectorSpaceSummary 9 1.2 ScalarorDotProduct 12 FreeMotionandOtherOrbits 14 1.3 VectororCrossProduct 20 1.4 TripleScalarProductandTripleVectorProduct 29 TripleScalarProduct 29 TripleVectorProduct 31 1.5 Gradient,∇ 35 PartialDerivatives 35 GradientasaVectorOperator 40 AGeometricalInterpretation 42 1.6 Divergence,∇ 44 APhysicalInterpretation 45 1.7 Curl,∇× 47 1.8 SuccessiveApplicationsof∇ 53 1.9 VectorIntegration 58 LineIntegrals 59 v vi Contents SurfaceIntegrals 62 VolumeIntegrals 65 IntegralDefinitionsofGradient,Divergence,andCurl 66 1.10 Gauss’sTheorem 68 Green’sTheorem 70 1.11 Stokes’sTheorem 72 1.12 PotentialTheory 76 ScalarPotential 76 1.13 Gauss’sLawandPoisson’sEquation 82 Gauss’sLaw 82 Poisson’sEquation 84 1.14 DiracDeltaFunction 86 AdditionalReading 95 2 VECTORANALYSISINCURVEDCOORDINATES ANDTENSORS 96 2.1 SpecialCoordinateSystems 97 RectangularCartesianCoordinates 97 IntegralsinCartesianCoordinates 98 2.2 CircularCylinderCoordinates 98 IntegralsinCylindricalCoordinates 101 Gradient 107 Divergence 108 Curl 110 2.3 OrthogonalCoordinates 113 2.4 DifferentialVectorOperators 121 Gradient 121 Divergence 122 Curl 124 2.5 SphericalPolarCoordinates 126 IntegralsinSphericalPolarCoordinates 130 2.6 TensorAnalysis 136 RotationofCoordinateAxes 137 InvarianceoftheScalarProductunderRotations 141 CovarianceofCrossProduct 142 CovarianceofGradient 143 Contents vii DefinitionofTensorsofRankTwo 144 AdditionandSubtractionofTensors 145 SummationConvention 145 Symmetry–Antisymmetry 146 Spinors 147 2.7 ContractionandDirectProduct 149 Contraction 149 DirectProduct 149 2.8 QuotientRule 151 2.9 DualTensors 153 Levi–CivitaSymbol 153 DualTensors 154 AdditionalReading 157 3 DETERMINANTSANDMATRICES 159 3.1 Determinants 159 LinearEquations:Examples 159 HomogeneousLinearEquations 160 InhomogeneousLinearEquations 161 LaplacianDevelopmentbyMinors 164 Antisymmetry 166 3.2 Matrices 174 BasicDefinitions,Equality,andRank 174 MatrixMultiplication,InnerProduct 175 DiracBra-ket,Transposition 178 Multiplication(byaScalar) 178 Addition 179 ProductTheorem 180 DirectProduct 182 DiagonalMatrices 182 Trace 184 MatrixInversion 184 3.3 OrthogonalMatrices 193 DirectionCosines 194 ApplicationstoVectors 195 OrthogonalityConditions:Two-DimensionalCase 198