Mathematical physics (Mathematical Methods for Physical Science)

544 Pages·2019·11.08 MB·English

by  Derek Raine

M
atheMatical
P
hysics
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M
atheMatical
P
hysics
An Introduction
Derek Raine, PhD et al.
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Original title and copyright: Mathematical Methods for Physical Science by
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CONTENTS
Preface ....................................................................................xiii
1 Derivatives and Integrals ............................................1
1.1 Definition of a derivative ...............................................1
1.2 Some basic derivatives ...................................................4
1.3 “Function of a function” (chain rule) ............................5
1.4 Product (and quotient) rule ...........................................8
1.5 Implicit differentiation .................................................11
1.6 Piecewise differentiable functions ...............................12
1.7 Higher order derivatives ..............................................14
1.8 Stationary points ...........................................................15
1.9 Integrals of elementary functions ................................18
1.10 Integrals of combinations of functions ........................20
1.11 Integration by substitution ...........................................21
1.12 Integration by parts ......................................................25
1.13 Integration of rational functions ..................................27
1.14 Definite integrals ..........................................................28
1.15 Area under a graph .......................................................29
1.16 Continuous and discontinuous functions ....................30
1.17 Estimates of integrals ...................................................32
1.18 Derivatives of integrals .................................................33
1.19 Reduction formulae ......................................................35
1.20 Exercises .......................................................................40
1.21 Problems .......................................................................44
vi • Mathematical Physics
2 Elementary Functions ...............................................49
2.1 Binomial expansion ......................................................50
2.2 Maclaurin series ...........................................................55
2.3 Taylor series ..................................................................58
2.4 Equilibrium points .......................................................59
2.5 Definition of ex (the exponential function) ..................63
2.6 The inverse function of ex.............................................63
2.7 Derivative of the exponential .......................................64
2.8 Integration of exponentials ..........................................67
2.9 Hyperbolic functions ....................................................68
2.10 Inverse functions ..........................................................69
2.11 Inverse hyperbolic functions ........................................71
2.12 Exercises .......................................................................72
2.13 Problems .......................................................................75
3 Functions, Limits, and Series ....................................77
3.1 Curve sketching: Quadratics ........................................77
3.2 Curve sketching: General polynomials ........................79
3.3 Curve sketching: Rational functions ............................81
3.4 Graphical solution of inequalities ................................83
3.5 The symmetry of functions ..........................................85
3.6 Limits ............................................................................87
3.7 Indeterminate forms ....................................................89
3.8 L’Hôpital’s rule .............................................................91
3.9 Limits of integrals .........................................................92
3.10 Approximation of functions ..........................................94
3.11 Sequences and series ....................................................96
3.12 Infinite series ................................................................97
3.13 Exercises .....................................................................104
3.14 Problems .....................................................................105
4 Vectors .....................................................................111
4.1 Basic properties ..........................................................112
4.2 Linear dependence; basis vectors ..............................114
4.3 The scalar (dot) product .............................................118
4.4 The vector (cross) product .........................................121
4.5 Multiple products .......................................................123
Contents • vii
4.6 Equation of a line .......................................................125
4.7 Equation of a plane ....................................................127
4.8 Components of vectors ..............................................129
4.9 Scalar (dot) product (component form) ....................131
4.10 Vector (cross) product (component form) .................132
4.11 Scalar triple product ...................................................133
4.12 Vector triple product ..................................................134
4.13 Vector identities ..........................................................135
4.14 Examples from physical science ................................138
4.15 Extension: Index notation ..........................................140
4.16 Exercises .....................................................................142
4.17 Problems .....................................................................142
5 Matrices ...................................................................147
5.1 Matrix representation .................................................149
5.2 Solution of systems of equations ................................153
5.3 Products ......................................................................156
5.4 The identity matrix .....................................................159
5.5 Symmetric and antisymmetric matrices ....................160
5.6 Inverses .......................................................................161
5.7 The inverse of a 2 × 2 matrix .....................................165
5.8 Determinants ..............................................................167
5.9 Properties of determinants ........................................169
5.10 Solution of 2 × 2 linear systems .................................174
5.11 Solution of 3 × 3 systems............................................177
5.12 Homogeneous systems ...............................................179
5.13 A formula for the inverse matrix ................................179
5.14 Eigenvalues and Eigenvectors ...................................181
5.15 Matrices as transformations .......................................186
5.16 Extension: Transformation groups .............................191
5.17 Exercises .....................................................................192
5.18 Problems .....................................................................195
6 Differential Equations 1 .........................................199
6.1 What are differential equations?................................199
6.2 Solving differential equations ....................................201
6.3 First order separable equations .................................203
viii • Mathematical Physics
6.4 Linearity versus non-linearity ....................................205
6.5 Homogeneous versus inhomogeneous ......................207
6.6 Finding solutions ........................................................209
6.7 Homogeneous linear equations .................................211
6.8 Auxiliary equations with repeated roots ....................215
6.9 Particular integrals .....................................................217
6.10 Inhomogeneous linear equations ...............................222
6.11 Integrating factor method ..........................................226
6.12 Extension: Special functions ......................................230
7 Complex Numbers ...................................................235
7.1 Introduction of complex numbers .............................235
7.2 Operations on complex numbers ...............................239
7.3 Quadratic equations ...................................................244
7.4 The Argand diagram ...................................................247
7.5 Modulus and argument of a complex number ..........249
7.6 The complex exponential ...........................................252
7.7 De Moivre’s theorem..................................................256
7.8 The roots of unity .......................................................258
7.9 Roots of real polynomials ...........................................258
7.10 Roots of complex polynomials ...................................259
7.11 Extension: Complex variable .....................................260
7.12 Exercises .....................................................................263
7.13 Problems .....................................................................264
8 Differential Equations 2 .........................................267
8.1 Boundary and initial conditions .................................267
8.2 Auxiliary equations with complex roots .....................277
8.3 Equations with complex coefficients .........................281
8.4 Complex inhomogeneous term ..................................282
8.5 Boundary (or initial) conditions .................................289
8.6 Systems of first order equations .................................291
8.7 Complex impedance ...................................................292
8.8 Some examples from physics .....................................294
8.9 Extension: Green functions .......................................296
8.10 Exercises .....................................................................298
8.11 Problems .....................................................................300
Contents • ix
9 Multiple Integrals ....................................................303
9.1 Repeated integrals ......................................................303
9.2 Integrals over rectangles in the plane ........................307
9.3 Multiple integrals over irregular regions ...................310
9.4 Change of order of integration ..................................313
9.5 Polar coordinates in two dimensions .........................316
9.6 Integrals over regions in the plane ............................317
9.7 An important definite integral ...................................319
9.8 Cylindrical polar coordinates .....................................320
9.9 Volume integrals (cylindrical polar) ...........................322
9.10 Integrals over cylindrical surfaces..............................323
9.11 Spherical polar coordinates ........................................325
9.12 Volume integrals (spherical polar) .............................326
9.13 Integrals over spherical surfaces ................................327
9.14 Solid angle ..................................................................329
9.15 Sketching surfaces ......................................................333
9.16 Exercises .....................................................................336
9.17 Problems .....................................................................338
10 Partial Derivatives .................................................341
10.1 Functions of two variables .........................................341
10.2 Partial derivatives .......................................................344
10.3 Directional derivatives ...............................................346
10.4 Functions of many variables ......................................348
10.5 Higher derivatives ......................................................349
10.6 Function of a function ................................................350
10.7 Implicit differentiation ...............................................352
10.8 Derivative with respect to a parameter .....................353
10.9 Taylor expansion about the origin ..............................355
10.10 Expansion about an arbitrary point ...........................357
10.11 Function of more than two variables .........................359
10.12 Stationary points .........................................................359
10.13 Definition of the total differential .............................365
10.14 Exact differentials .......................................................366
10.15 The chain rule .............................................................367
10.16 Consistency with the chain rule .................................369
10.17 Exercises .....................................................................372
10.18 Problems .....................................................................375
x • Mathematical Physics
11 Partial Differential Equations ..............................379
11.1 Examples from physics ...............................................380
11.2 General solution of the wave equation ......................383
11.3 Derivation of the general solution .............................386
11.4 A string initially at rest ...............................................388
11.5 A string given an initial velocity .................................391
11.6 A formula for general initial conditions .....................393
11.7 Semi-infinite string .....................................................395
11.8 Simple harmonic waves ..............................................397
11.9 Beats............................................................................397
11.10 Group velocity ............................................................400
11.11 Separation of variables ...............................................402
11.12 Eigenvalues .................................................................406
11.13 Reflections at a discontinuity .....................................408
11.14 Matter waves incident on a potential step .................412
11.15 Exercises .....................................................................415
11.16 Problems .....................................................................416
12 Fourier Series ........................................................419
12.1 Introduction ................................................................419
12.2 Fourier half range sine series .....................................420
12.3 Fourier sine series coefficients ..................................421
12.4 Using the Fourier series results .................................422
12.5 Application to the wave equation ..............................425
12.6 Application to other differential equations ...............428
12.7 Fourier half range cosine series .................................430
12.8 Numerical series .........................................................432
12.9 Periodic extension of Fourier series ..........................433
12.10 Even and odd symmetry of periodic functions .........433
12.11 Fourier full range series .............................................435
12.12 Complex form of Fourier series .................................436
12.13 Properties of Fourier series .......................................439
12.14 Fourier transforms and Fourier integrals ..................440
12.15 Fourier transforms .....................................................441
12.16 Calculating Fourier transforms ..................................442
12.17 Convolutions ...............................................................443
12.18 Convolution theorem .................................................444