Essential Mathematical Methods for Physicists: and Engineers

959 Pages·2003·5.75 MB·English

by  Hans J. Weber

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
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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