Mathematical Physics with Partial Differential Equations

Front Cover......Page 1
Title Page\n......Page 5
Copyright Page......Page 6
Contents......Page 7
Preface......Page 13
1-1 SELF-ADJOINT OPERATORS......Page 15
Fourier Coefficients......Page 19
Exercises......Page 25
1-2 CURVILINEAR COORDINATES......Page 28
Scaling Factors......Page 31
Volume Integrals......Page 32
The Gradient......Page 36
The Laplacian......Page 37
Other Curvilinear Systems......Page 39
Applications......Page 45
An Alternate Approach (Optional)......Page 46
Exercises......Page 47
1-3 APPROXIMATE IDENTITIES AND THE DIRAC-δ FUNCTION......Page 48
Approximate Identities......Page 49
The Dirac-δ Function in Physics......Page 51
Some Calculus for the Dirac-δ Function......Page 54
The Dirac-δ Function in Curvilinear Coordinates......Page 56
Exercises......Page 58
Series of Real Numbers......Page 59
Convergence versus Absolute Convergence......Page 61
Series of Functions......Page 62
Power Series......Page 68
Taylor Series......Page 70
Exercises......Page 74
1-5 SOME IMPORTANT INTEGRATION FORMULAS......Page 78
Other Facts We Will Use Later......Page 82
Another Important Integral......Page 83
Exercises......Page 84
2-1 Vector Integration......Page 87
Path Integrals......Page 88
Line Integrals......Page 91
Surfaces......Page 94
Parameterized Surfaces......Page 96
Integrals of Scalar Functions Over Surfaces......Page 97
Surface Integrals of Vector Functions......Page 99
Exercises......Page 105
Line Integrals......Page 106
2-2 Divergence and Curl......Page 107
Cartesian Coordinate Case......Page 108
Cylindrical Coordinate Case......Page 111
Spherical Coordinate Case......Page 114
The Curl in Cartesian Coordinates......Page 118
The Curl in Cylindrical Coordinates......Page 123
The Curl in Spherical Coordinates......Page 128
2-3 Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem......Page 136
The Divergence (Gauss’) Theorem......Page 141
Stokes’ Theorem......Page 149
An Application of Stokes’ Theorem......Page 154
An Application of the Divergence Theorem......Page 155
Conservative Fields......Page 156
Green’s Theorem Problems......Page 162
Stokes’ Theorem Problems......Page 163
Divergence Theorem Problems......Page 166
Conservative Field Problems......Page 167
Introduction......Page 169
3-1 Construction of Green’s Function Using the Dirac-δ Function......Page 170
3-2 Construction of Green’s Function Using Variation of Parameters......Page 178
3-3 Construction of Green’s Function from Eigenfunctions......Page 182
3-4 More General Boundary Conditions......Page 185
3-5 The Fredholm Alternative (or, What If 0 is an Eigenvalue?)......Page 187
3-6 Green’s function for the Laplacian in higher dimensions......Page 194
Exercises......Page 200
Introduction......Page 201
4-1 Basic Definitions......Page 202
Exercises......Page 205
4-2 Methods of Convergence of Fourier Series......Page 207
Fourier Series on Arbitrary Intervals......Page 213
Exercises......Page 218
4-3 The Exponential Form of Fourier Series......Page 220
Exercises......Page 221
4-4 Fourier Sine and Cosine Series......Page 222
4-5 Double Fourier Series......Page 224
Exercise......Page 226
Introduction......Page 227
5-1 Laplace’s Equation......Page 229
5-2 Derivation of the Heat Equation in One Dimension......Page 230
5-3 Derivation of the Wave equation in One Dimension......Page 232
5-4 An Explicit Solution of the Wave Equation......Page 236
Exercises......Page 241
5-5 Converting Second-order PDEs to Standard Form......Page 242
Exercise......Page 246
Introduction......Page 247
6-1 The Self-Adjoint Property of a Sturm-Liouville Equation......Page 248
Exercises......Page 250
6-2 Completeness of Eigenfunctions for Sturm-Liouville Equations......Page 251
6-3 Uniform Convergence of Fourier Series......Page 259
7-1 Solving Laplace’s Equation on a Rectangle......Page 265
Exercises......Page 270
7-2 Laplace’s Equation on a Cube......Page 272
Exercises......Page 275
7-3 Solving the Wave Equation in One Dimension by Separation of Variables......Page 276
Exercises......Page 281
7-4 Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables......Page 283
7-5 Solving the heat equation in one dimension using separation of variables......Page 285
The Initial Condition Is the Dirac-δ Function......Page 288
Exercises......Page 290
7-6 Steady State of the Heat equation......Page 291
Exercises......Page 295
7-7 Checking the Validity of the Solution......Page 297
An Example Where Bessel Functions Arise......Page 301
8-1 The Solution to Bessel’s Equation in Cylindrical Coordinates......Page 306
Exercises......Page 308
8-2 Solving Laplace’s Equation in Cylindrical Coordinates Using Separation of Variables......Page 309
8-3 The Wave Equation on a Disk (Drum Head Problem)......Page 313
8-4 The Heat Equation on a Disk......Page 317
9-1 An Example Where Legendre Equations Arise......Page 321
9-2 The Solution to Bessel’s Equation in Spherical Coordinates......Page 324
9-3 Legendre’s Equation and Its Solutions......Page 329
Exercises......Page 332
9-4 Associated Legendre Functions......Page 333
9-5 Laplace’s Equation in Spherical Coordinates......Page 336
Exercise......Page 339
Introduction......Page 341
10-1 The Fourier Transform as a Decomposition......Page 342
10-2 The Fourier Transform from the Fourier Series......Page 343
10-3 Some Properties of the Fourier Transform......Page 345
Exercises......Page 348
10-4 Solving Partial Differential Equations Using the Fourier Transform......Page 349
Exercises......Page 355
10-5 The Spectrum of the Negative Laplacian in One Dimension......Page 357
10-6 The Fourier Transform in Three Dimensions......Page 360
Exercise......Page 364
Introduction......Page 365
11-1 Properties of the Laplace Transform......Page 366
11-2 Solving Differential Equations Using the Laplace Transform......Page 370
Exercises......Page 374
11-3 Solving the Heat Equation using the Laplace Transform......Page 375
Exercises......Page 380
11-4 The Wave Equation and the Laplace Transform......Page 382
Exercises......Page 387
12-1 Solving the Heat Equation Using Green’s Function......Page 389
Green’s Function for the Nonhomogeneous Heat Equation......Page 391
Method of Images for a Semi-infinite Interval......Page 393
Method of Images for a Bounded Interval......Page 397
Exercises......Page 403
12-3 Green’s Function for the Wave Equation......Page 404
Exercises......Page 411
12-4 Green’s Function and Poisson’s Equation......Page 412
Exercises......Page 415
CYLINDRICAL COORDINATES......Page 417
THE LAPLACIAN IN SPHERICAL COORDINATES......Page 422
References......Page 427
Index......Page 429