Preface ........................................................ xi
1 Preliminaries ................................................ 1
1-1 Self-Adjoint Operators .................................. 1
Fourier Coefficients ................................... 5
Exercises .............................................. 11
1-2 Curvilinear Coordinates ............................... 14
Scaling Factors ........................................ 17
Volume Integrals ...................................... 18
The Gradient ........................................... 22
The Laplacian .......................................... 23
Spherical Coordinates .................................. 25
Other Curvilinear Systems ............................. 25
Applications ........................................... 31
An Alternate Approach (Optional) ....................... 33
Exercises .............................................. 33
1-3 Approximate Identities and the Dirac-6 Function ........ 34
Approximate Identities ................................. 35
The Dirac-δ Function in Physics ........................ 37
Some Calculus for the Dirac-δ Function ................. 40
The Dirac-δ Function in Curvilinear Coordinates ........ 42
Exercises .............................................. 44
1-4 The Issue of Convergence ............................... 45
Series of Real Numbers ................................. 45
Convergence versus Absolute Convergence ................ 47
Series of Functions .................................... 48
Power Series ........................................... 54
Taylor Series .......................................... 56
Exercises .............................................. 60
1-5 Some Important Integration Formulas .................... 64
Other Facts We Will Use Later .......................... 68
Another Important Integral ............................. 69
Exercises .............................................. 74
2 Vector Calculus ............................................. 73
2-1 Vector Integration ..................................... 73
Path Integrals ......................................... 74
Line Integrals ......................................... 77
Surfaces ............................................... 80
Parameterized Surfaces ................................. 82
Integrals of Scalar Functions Over Surfaces ............ 83
Surface Integrals of Vector Functions .................. 85
Exercises .............................................. 91
2-2 Divergence and Curl .................................... 93
Cartesian Coordinate Case .............................. 94
Cylindrical Coordinate Case ............................ 97
Spherical Coordinate Case ............................. 100
The Curl .............................................. 104
The Curl in Cartesian Coordinates ..................... 104
The Curl in Cylindrical Coordinates ................... 109
The Curl in Spherical Coordinates ..................... 114
Exercises ............................................. 122
2-3 Green's Theorem, the Divergence Theorem, and
Stokes'Theorem ........................................ 122
The Divergence (Gauss') Theorem ....................... 127
Stokes' Theorem ....................................... 135
An Application of Stokes' Theorem ..................... 140
An Application of the Divergence Theorem .............. 141
Conservative Fields ................................... 142
Exercises ............................................. 148
3 Green's Functions .......................................... 155
Introduction ............................................... 155
3-1 Construction of Green's Function Using the Dirac-δ
Function .............................................. 156
Exercises ............................................. 164
3-2 Construction of Green's Function Using Variation of
Parameters ............................................ 164
Exercises ............................................. 168
3-3 Construction of Green's Function from Eigenfunctions .. 168
Exercises ............................................. 171
3-4 More General Boundary Conditions ...................... 171
Exercises ............................................. 173
3-5 The Fredholm Alternative (or, What If 0 Is an
Eigenvalue?) .......................................... 173
Exercises .. 180
3-6 Green's Function for the Laplacian in Higher
Dimensions ............................................ 180
Exercises ............................................. 186
4 Fourier Series ............................................. 187
Introduction ............................................... 187
4-1 Basic Definitions ..................................... 188
Exercises ............................................. 191
4-2 Methods of Convergence of Fourier Series .............. 193
Fourier Series on Arbitrary Intervals ................. 203
Exercises ............................................. 204
4-3 The Exponential Form of Fourier Series ................ 206
Exercises ............................................. 207
4-4 Fourier Sine and Cosine Series ........................ 208
Exercises ............................................. 210
4-5 Double Fourier Series ................................. 210
Exercise .............................................. 212
5 Three Important Equations .................................. 213
Introduction ............................................... 213
5-1 Laplace's Equation .................................... 215
Exercises ............................................. 216
5-2 Derivation of the Heat Equation in One Dimension ...... 216
Exercise .............................................. 218
5-3 Derivation of the Wave Equation in One Dimension ...... 218
Exercises ............................................. 222
5-4 An Explicit Solution of the Wave Equation ............. 222
Exercises ............................................. 227
5-5 Converting Second-Order PDEs to Standard Form ......... 228
Exercise ................................................... 232
6 Sturm-Liouville Theory ..................................... 233
Introduction ............................................... 233
Exercises .................................................. 234
6-1 The Self-Adjoint Property of a Sturm-Liouville
Equation .............................................. 234
Exercises ............................................. 236
6-2 Completeness of Eigenfunctions for Sturm-Liouville
Equations ............................................. 237
Exercises ............................................. 245
6-3 Uniform Convergence of Fourier Series ................. 245
7 Separation of Variables in Cartesian Coordinates ........... 251
Introduction ............................................... 251
7-1 Solving Laplace's Equation on a Rectangle ............. 251
Exercises ............................................. 256
7-2 Laplace's Equation on a Cube .......................... 258
Exercises ............................................. 261
7-3 Solving the Wave Equation in One Dimension by
Separation of Variables ............................... 262
Exercises ............................................. 267
7-4 Solving the Wave Equation in Two Dimensions in
Cartesian Coordinates by Separation of Variables ...... 269
Exercises ............................................. 271
7-5 Solving the Heat Equation in One Dimension Using
Separation of Variables ............................... 271
The Initial Condition Is the Dirac-6 Function ......... 274
Exercises ............................................. 276
7-6 Steady State of the Heat Equation ..................... 277
Exercises ............................................. 281
7-7 Checking the Validity of the Solution ................. 283
8 Solving Partial Differential Equations in Cylindrical
Coordinates Using Separation of Variables .................. 287
Introduction ............................................... 287
An Example Where Bessel Functions Arise .................... 287
Exercises .................................................. 292
8-1 The Solution to Bessel's Equation in Cylindrical
Coordinates ........................................... 292
Exercises ............................................. 294
8-2 Solving Laplace's Equation in Cylindrical Coordinates
Using Separation of Variables ......................... 295
Exercises ............................................. 299
8-3 The Wave Equation on a Disk (Drum Head Problem) ....... 299
Exercises ............................................. 303
8-4 The Heat Equation on a Disk ........................... 303
Exercises ............................................. 306
9 Solving Partial Differential Equations in Spherical
Coordinates Using Separation of Variables .................. 307
9-1 An Example Where Legendre Equations Arise ............. 307
9-2 The Solution to Bessel's Equation in Spherical
Coordinates ........................................... 310
9-3 Legendre's Equation and Its Solutions ................. 315
Exercises ............................................. 318
9-4 Associated Legendre Functions ......................... 319
Exercise .............................................. 322
9-5 Laplace's Equation in Spherical Coordinates ........... 322
Exercise .............................................. 325
10 The Fourier Transform ...................................... 327
Introduction ............................................... 327
10-1 The Fourier Transform as a Decomposition .............. 328
10-2 The Fourier Transform from the Fourier Series ......... 329
10-3 Some Properties of the Fourier Transform .............. 331
Exercises ............................................. 334
10-4 Solving Partial Differential Equations Using the
Fourier Transform ..................................... 335
Exercises ............................................. 341
10-5 The Spectrum of the Negative Laplacian in One
Dimension ............................................. 343
10-6 The Fourier Transform in Three Dimensions ............. 346
Exercise .............................................. 350
11 The Laplace Transform ...................................... 351
Introduction ............................................... 351
Exercises .................................................. 352
11-1 Properties of the Laplace Transform ................... 352
Exercises ............................................. 356
11-2 SolvThg Differential Equations Using the Laplace
Transform ............................................. 356
Exercises ............................................. 360
11-3 Solving the Heat Equation Using the Laplace Transform . 361
Exercises ............................................. 366
11-4 The Wave Equation and the Laplace Transform ........... 368
Exercises ............................................. 373
12 Solving PDEs with Green's Functions ........................ 375
12-1 Solving the Heat Equation Using Green's Function ...... 375
Green's Function for the Nonhomogeneous Heat
Equation .............................................. 377
Exercises ............................................. 379
12-2 The Method of Images .................................. 379
Method of Images for a Semi-infinite Interval ......... 379
Method of Images for a Bounded Interval ............... 383
Exercises ............................................. 389
12-3 Green's Function for the Wave Equation ................ 390
Exercises ............................................. 397
12-4 Green's Function and Poisson's Equation ............... 398
Exercises ............................................. 401
Appendix: Computing the Laplacian with the Chain Rule ......... 403
References .................................................... 413
Index ......................................................... 415
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