Mathematical methods for physics and engineering

736 Pages·2018·14.019 MB·English

by  Blennow, Mattias, Physik

Mathematical Methods
for Physics
and Engineering

Mathematical Methods
for Physics
and Engineering
Mattias Blennow
CRC Press
Taylor & Francis Group
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Version Date: 20171121
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Contents
hapter
C 1(cid:4) Scalars and Vectors 1
1.1 VECTORSANDARITHMETICS 1
1.2 ROTATIONSANDBASISCHANGES 3
1.3 INDEXNOTATION 6
1.3.1 The Kronecker delta and the permutation symbol 6
1.3.2 Vector algebra using index notation 7
1.4 FIELDS 10
1.4.1 Locality 12
1.4.2 Field integrals 13
1.4.2.1 Volume integrals 13
1.4.2.2 Surface integrals 14
1.4.2.3 Line integrals 16
1.4.3 Di(cid:11)erential operators and (cid:12)elds 17
1.4.3.1 The gradient 18
1.4.3.2 The divergence 19
1.4.3.3 The curl 20
1.4.3.4 The directional derivative 22
1.4.3.5 Second order operators 22
1.4.3.6 Coordinate independence 23
1.5 INTEGRALTHEOREMS 24
1.5.1 Line integral of a gradient 24
1.5.2 The divergence theorem 25
1.5.3 Green’s formula 29
1.5.4 The curl theorem 29
1.5.5 General integral theorems 31
1.6 NON-CARTESIANCOORDINATESYSTEMS 32
1.6.1 General theory 32
1.6.1.1 Tangent vector basis 34
1.6.1.2 Dual basis 35
1.6.2 Orthogonal coordinates 38
1.6.2.1 Integration in orthogonal coordinates 39
1.6.2.2 Di(cid:11)erentiation in orthogonal coordinates 40
vii
viii (cid:4) Contents
1.6.3 Polar and cylinder coordinates 42
1.6.4 Spherical coordinates 44
1.7 POTENTIALS 49
1.7.1 Scalar potentials 49
1.7.2 Vector potentials 53
1.7.3 Scalar and vector potentials 57
1.8 PROBLEMS 59
hapter
C 2(cid:4) Tensors 69
2.1 OUTERPRODUCTSANDTENSORBASES 71
2.1.1 General coordinate bases 72
2.2 TENSORALGEBRA 74
2.2.1 Tensors and symmetries 75
2.2.2 The quotient law 78
2.3 TENSORFIELDSANDDERIVATIVES 79
2.3.1 The metric tensor 80
2.3.1.1 Distances and the metric tensor 81
2.3.1.2 Lowering and raising indices 82
2.3.2 Derivatives of tensor (cid:12)elds 83
2.3.2.1 The covariant derivative 85
2.3.2.2 Divergence 87
2.3.2.3 Generalised curl 88
2.3.3 Tensor densities 89
2.3.4 The generalised Kronecker delta 93
2.3.5 Orthogonal coordinates 95
2.4 TENSORSINCARTESIANCOORDINATES 96
2.5 TENSORINTEGRALS 98
2.5.1 Integration of tensors in Cartesian coordinates 98
2.5.1.1 Volume integration 98
2.5.1.2 Surface integrals 99
2.5.1.3 Line integrals 100
2.5.1.4 Integral theorems 101
2.5.2 The volume element and general coordinates 102
2.6 TENSOREXAMPLES 104
2.6.1 Solid mechanics 105
2.6.1.1 The stress tensor 105
2.6.1.2 The strain tensor 107
2.6.1.3 The sti(cid:11)ness and compliance tensors 109
2.6.2 Electromagnetism 110
2.6.2.1 The magnetic (cid:12)eld tensor 111
Contents (cid:4) ix
2.6.2.2 The Maxwell stress tensor 111
2.6.2.3 The conductivity and resistivity tensors 113
2.6.3 Classical mechanics 115
2.6.3.1 The moment of inertia tensor 115
2.6.3.2 The generalised inertia tensor 117
2.7 PROBLEMS 119
hapter
C 3(cid:4) Partial Differential Equations and Modelling 127
3.1 AQUICKNOTEONNOTATION 127
3.2 INTENSIVEANDEXTENSIVEPROPERTIES 128
3.3 THECONTINUITYEQUATION 130
3.4 THEDIFFUSIONANDHEATEQUATIONS 134
3.4.1 Di(cid:11)usion and Fick’s laws 134
3.4.2 Heat conduction and Fourier’s law 136
3.4.3 Additional convection currents 137
3.5 THEWAVEEQUATION 138
3.5.1 Transversal waves on a string 139
3.5.1.1 Wave equation as an application of continuity 140
3.5.2 Transversal waves on a membrane 141
3.5.3 Electromagnetic waves 143
3.6 BOUNDARYANDINITIALCONDITIONS 144
3.6.1 Boundary conditions 145
3.6.1.1 Dirichlet conditions 145
3.6.1.2 Neumann conditions 146
3.6.1.3 Robin boundary conditions 147
3.6.2 Initial conditions 148
3.6.3 Uniqueness 150
3.7 PDESINSPACEONLY 151
3.8 LINEARISATION 154
3.9 THECAUCHYMOMENTUMEQUATIONS 157
3.9.1 Inviscid (cid:13)uids 159
3.9.2 Navier{Stokes equations 161
3.9.3 Incompressible (cid:13)ow 164
3.10 SUPERPOSITIONANDINHOMOGENEITIES 165
3.10.1 Removing inhomogeneities from boundaries 166
3.10.2 Using known solutions 167
3.11 MODELLINGTHINVOLUMES 168
3.12 DIMENSIONALANALYSIS 170
3.12.1 Units 172
3.12.2 The Buckingham (cid:25) theorem 174