Mathematical tools for physics

Nearing J.

Table of contents :
CONTENTS……Page 2
INTRODUCTION……Page 5
BIBLIOGRAPHY……Page 7
Radians……Page 9
Hyperbolic Functions……Page 10
2.Parametric Differentiation……Page 13
3.Gaussian Integrals……Page 14
4.erf and Gamma……Page 16
Why erf?……Page 18
5.Differentiating……Page 19
6.Integrals……Page 20
Fundamental Thm. of Calculus……Page 22
Riemann-Stieljes Integrals……Page 23
Yes……Page 24
Partial Integration……Page 27
7.Polar Coordinates……Page 28
8.Sketching Graphs……Page 29
Problems……Page 32
1.The Basics……Page 38
2.Deriving Taylor Series……Page 40
Ratio Test……Page 42
Integral Test……Page 43
Quicker Comparison Test……Page 44
Absolute Convergence……Page 45
4.Series of Series……Page 46
5.Power series, two variables……Page 47
6.Stirling’s Approximation……Page 49
Probability Distribution……Page 50
7.Useful Tricks……Page 53
8.Diffraction……Page 54
9.Checking Results……Page 58
Electrostatics Example……Page 60
Estimating a tough integral……Page 62
Problems……Page 64
1.Complex Numbers……Page 73
2.Some Functions……Page 74
Complex Exponentials……Page 76
3.Applications of Euler’s Formula……Page 77
Complex Conjugate……Page 79
Roots of Unity……Page 80
4.Logarithms……Page 81
5.Mapping……Page 82
Problems……Page 84
1.Linear Constant-Coefficient……Page 91
Damped Oscillator……Page 94
2.Forced Oscillations……Page 95
Regular Singular Points……Page 99
4.Trigonometry via ODE’s……Page 103
5.Green’s Functions……Page 104
6.Separation of Variables……Page 107
7.Simultaneous Equations……Page 109
8.Simultaneous ODE’s……Page 111
9.Legendre’s Equation……Page 115
Problems……Page 118
1.Examples……Page 126
2.Computing Fourier Series……Page 127
More Examples……Page 128
3.Choice of Basis……Page 130
Fundamental Theorem……Page 131
Parseval’s Identity……Page 135
4.Periodically Forced ODE’s……Page 136
Pure Frequency Forcing……Page 137
General Periodic Force……Page 138
5.Return to Parseval……Page 140
Other Applications……Page 141
6.Gibbs Phenomenon……Page 142
Problems……Page 144
1.The Underlying Idea……Page 150
2.Axioms……Page 151
3.Examples of Vector Spaces……Page 152
Special Function Space……Page 155
Bases, Dimension, Components……Page 156
Differential Equations……Page 158
Examples……Page 160
7.Bases and Scalar Products……Page 163
9.Cauchy-Schwartz inequality……Page 164
Norm from a Scalar Product……Page 165
10.Infinite Dimensions……Page 166
Problems……Page 168
1.The Idea of an Operator……Page 176
2.Definition of an Operator……Page 180
3.Examples of Operators……Page 181
Components of Rotations……Page 183
Components of Inertia……Page 184
Components of Dumbbell……Page 185
Parallel Axis Theorem……Page 187
Components of the Derivative……Page 189
4.Matrix Multiplication……Page 190
5.Inverses……Page 191
6.Areas, Volumes, Determinants……Page 193
7.Matrices as Operators……Page 194
Determinant of Composition……Page 195
8.Eigenvalues and Eigenvectors……Page 196
Example of Eigenvectors……Page 198
Example: Coupled Oscillators……Page 199
9.Change of Basis……Page 200
Similarity Transformations……Page 202
Eigenvectors……Page 203
11.Can you Diagonalize a Matrix?……Page 204
Differential Equations at Critical……Page 205
12.Eigenvalues and Google……Page 207
Problems……Page 209
1.Partial Derivatives……Page 216
2.Differentials……Page 217
Differentials in Several Variables……Page 218
3.Chain Rule……Page 219
4.Geometric Interpretation……Page 222
Examples……Page 223
5.Gradient……Page 224
6.Electrostatics……Page 225
Vibrating Drumhead……Page 227
8.Cylindrical, Spherical Coordinates……Page 228
Examples of Multiple Integrals……Page 229
An Area……Page 230
A Surface Charge Density……Page 231
9.Vectors: Cylindrical, Spherical Bases……Page 232
Nuclear Magnetic Field……Page 233
10.Gradient in other Coordinates……Page 234
11.Maxima, Minima, Saddles……Page 235
12.Lagrange Multipliers……Page 237
Examples of Lagrange Multipliers……Page 238
13.Solid Angle……Page 241
Cross Section, Absorption……Page 242
Cross Section, Scattering……Page 243
14.Rainbow……Page 244
15.3D Visualization……Page 248
Problems……Page 249
1.Fluid Flow……Page 256
General Flow, Curved Surfaces……Page 257
Example of Flow Calculation……Page 258
Another Flow Calculation……Page 259
2.Vector Derivatives……Page 261
Div, Curl, Strain……Page 262
3.Computing the divergence……Page 263
The Divergence as Derivatives……Page 265
Simplifying the derivation……Page 267
4.Integral Representation of Curl……Page 269
The Curl in Components……Page 270
5.The Gradient……Page 271
6.Shorter Cut for div and curl……Page 272
7.Identities for Vector Operators……Page 273
8.Applications to Gravity……Page 274
9.Gravitational Potential……Page 276
Boundary Conditions……Page 277
Back to the Problem……Page 278
10.Summation Convention……Page 279
11.More Complicated Potentials……Page 280
Problems……Page 282
1.The Heat Equation……Page 291
In Three Dimensions……Page 292
2.Separation of Variables……Page 293
Example……Page 295
3.Oscillating Temperatures……Page 296
4.Spatial Temperature Distributions……Page 298
The Heat Flow into the Box……Page 303
5.Specified Heat Flow……Page 305
6.Electrostatics……Page 308
More Electrostatic Examples……Page 313
Problems……Page 316
1.Interpolation……Page 323
2.Solving equations……Page 326
3.Differentiation……Page 328
4.Integration……Page 332
Simpson’s Rule……Page 334
Gaussian Integration……Page 336
5.Differential Equations……Page 337
Runge-Kutta……Page 338
Adams Methods……Page 340
Instability……Page 342
6.Fitting of Data……Page 344
7.Euclidean Fit……Page 346
Correlation, Principal Components……Page 349
8.Differentiating noisy data……Page 350
9.Partial Differential Equations……Page 352
Problems……Page 356
1.Examples……Page 362
Definition of “Function”……Page 363
Functional……Page 364
Multilinear Functionals……Page 366
2.Components……Page 368
Change of Basis……Page 371
Change of Basis (more efficient)……Page 374
3.Relations between Tensors……Page 376
Symmetries……Page 378
Alternating Tensor……Page 379
4.Non-Orthogonal Bases……Page 380
Reciprocal Basis……Page 381
Summation Convention……Page 382
Metric Tensor……Page 384
Raising and Lowering……Page 386
5.Manifolds and Fields……Page 387
6.Coordinate Systems……Page 389
Coordinate Basis……Page 390
Reciprocal Coordinate Basis……Page 391
Example……Page 393
7.Basis Change……Page 394
Problems……Page 400
1.Integrals……Page 404
Weighted Integrals……Page 406
2.Line Integrals……Page 407
3.Gauss’s Theorem……Page 409
4.Stokes’ Theorem……Page 410
Example……Page 413
Conservative Fields……Page 414
Potentials……Page 415
Vector Potentials……Page 416
5.Reynolds’ Transport Theorem……Page 417
Faraday’s Law……Page 419
Problems……Page 420
1.Differentiation……Page 426
2.Integration……Page 429
3.Power (Laurent) Series……Page 431
5.Branch Points……Page 434
Example 2……Page 435
Example 3……Page 436
Example 4……Page 437
Example 5……Page 439
Example 6……Page 440
Example 7……Page 441
7.Branch Points……Page 443
Geometry of Branch Points……Page 444
Other Functions……Page 446
Example 8……Page 448
9.Other Results……Page 450
Problems……Page 453
1.Fourier Transform……Page 459
Examples……Page 461
2.Convolution Theorem……Page 463
Example……Page 464
Example……Page 465
4.Derivatives……Page 466
5.Green’s Functions……Page 467
6.Sine and Cosine Transforms……Page 470
7.Weiner-Khinchine Theorem……Page 472
Problems……Page 473
INDEX……Page 477