Mathematics for physics and physicists

Title:

Personal Author:

Appel, Walter

Publication Information:

Princeton, NJ : Princeton University Press, 2007

ISBN:

9780691131023

Subject Term:

Mathematical physics

Mathematical physics -- Problems, exercises, etc.

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010155364	QC20 A66 2001	Open Access Book	Book	Searching... Unknown

What can a physicist gain by studying mathematics? By gathering together everything a physicist needs to know about mathematics in one comprehensive and accessible guide, this is the question Mathematics for Physics and Physicists successfully takes on.

The author, Walter Appel, is a renowned mathematics educator hailing from one of the best schools of France's prestigious Grandes écoles, where he has taught some of his country's leading scientists and engineers. In this unique book, oriented specifically toward physicists, Appel shows graduate students and researchers the vital benefits of integrating mathematics into their study and experience of the physical world. His approach is mathematically rigorous yet refreshingly straightforward, teaching all the math a physicist needs to know above the undergraduate level. Appel details numerous topics from the frontiers of modern physics and mathematics--such as convergence, Green functions, complex analysis, Fourier series and Fourier transform, tensors, and probability theory--consistently partnering clear explanations with cogent examples. For every mathematical concept presented, the relevant physical application is discussed, and exercises are provided to help readers quickly familiarize themselves with a wide array of mathematical tools.

Mathematics for Physics and Physicists is the resource today's physicists must have to strengthen their math skills and to gain otherwise unattainable insights into their fields of study.

Author Notes

Walter Appel taught mathematics for physics for seven years at the Ecole Normale Superieure de Lyon in France and currently teaches mathematics at the Henri Poincare School

Reviews 1

Choice Review

The majority of applied mathematical fields presently require so much specialization that mathematics often takes a back seat to the particular field of study. This book not only contains a great deal of the mathematics necessary to seriously study physics but also encourages physicists and potential physicists to embrace mathematics. From knowing mathematics, physicists "will gain access to new intuitions." Many great physicists, including Galileo, Newton, and Feynman, are excellent examples of those who gained immense insight from the study of mathematics, which enabled them to obtain important results in physics. Mathematical topics here include calculus, measure theory, complex analysis, distributions, Fourier and Laplace transforms, group representations, and probability theory. Appel (Henri Poincare School) compiled this book based on his experience of some seven years teaching mathematics to physicists. His underlying premise is that in order for physicists to attain insights into solving physics problems, they must continue to strengthen their mathematics skills. A comprehensive guide to mathematics essential to physics, the book is neither too imprecise nor too specialized to be used as a course resource. Summing Up: Recommended. Upper-division undergraduates through faculty. J. T. Zerger Catawba College

A book's apology	p. xviii
Index of notation	p. xxii
1 Reminders: convergence of sequences and series	p. 1
1.1 The problem of limits in physics	p. 1
1.1.a Two paradoxes involving kinetic energy	p. 1
1.1.b Romeo, Juliet, and viscous fluids	p. 5
1.1.c Potential wall in quantum mechanics	p. 7
1.1.d Semi-infinite filter behaving as waveguide	p. 9
1.2 Sequences	p. 12
1.2.a Sequences in a normed vector space	p. 12
1.2.b Cauchy sequences	p. 13
1.2.c The fixed point theorem	p. 15
1.2.d Double sequences	p. 16
1.2.e Sequential definition of the limit of a function	p. 17
1.2.f Sequences of functions	p. 18
1.3 Series	p. 23
1.3.a Series in a normed vector space	p. 23
1.3.b Doubly infinite series	p. 24
1.3.c Convergence of a double series	p. 25
1.3.d Conditionally convergent series, absolutely convergent series	p. 26
1.3.e Series of functions	p. 29
1.4 Power series, analytic functions	p. 30
1.4.a Taylor formulas	p. 31
1.4.b Some numerical illustrations	p. 32
1.4.c Radius of convergence of a power series	p. 34
1.4.d Analytic functions	p. 35
1.5 A quick look at asymptotic and divergent series	p. 37
1.5.a Asymptotic series	p. 37
1.5.b Divergent series and asymptotic expansions	p. 38
Exercises	p. 43
Problem	p. 46
Solutions	p. 47
2 Measure theory and the Lebesgue integral	p. 51
2.1 The integral according to Mr. Riemann	p. 51
2.1.a Riemann sums	p. 51
2.1.b Limitations of Riemann's definition	p. 54
2.2 The integral according to Mr. Lebesgue	p. 54
2.2.a Principle of the method	p. 55
2.2.b Borel subsets	p. 56
2.2.c Lebesgue measure	p. 58
2.2.d The Lebesgue [sigma]-algebra	p. 59
2.2.e Negligible sets	p. 61
2.2.f Lebesgue measure on R[superscript n]	p. 62
2.2.g Definition of the Lebesgue integral	p. 62
2.2.h Functions zero almost everywhere, space L[superscript 1]	p. 66
2.2.i And today?	p. 67
Exercises	p. 68
Solutions	p. 71
3 Integral calculus	p. 73
3.1 Integrability in practice	p. 73
3.1.a Standard functions	p. 73
3.1.b Comparison theorems	p. 74
3.2 Exchanging integrals and limits or series	p. 75
3.3 Integrals with parameters	p. 77
3.3.a Continuity of functions defined by integrals	p. 77
3.3.b Differentiating under the integral sign	p. 78
3.3.c Case of parameters appearing in the integration range	p. 78
3.4 Double and multiple integrals	p. 79
3.5 Change of variables	p. 81
Exercises	p. 83
Solutions	p. 85
4 Complex Analysis I	p. 87
4.1 Holomorphic functions	p. 87
4.1.a Definitions	p. 88
4.1.b Examples	p. 90
4.1.c The operators [part]/[part]z and [part]/[part]z	p. 91
4.2 Cauchy's theorem	p. 93
4.2.a Path integration	p. 93
4.2.b Integrals along a circle	p. 95
4.2.C Winding number	p. 96
4.2.d Various forms of Cauchy's theorem	p. 96
4.2.e Application	p. 99
4.3 Properties of holomorphic functions	p. 99
4.3.a The Cauchy formula and applications	p. 99
4.3.b Maximum modulus principle	p. 104
4.3.c Other theorems	p. 105
4.3.d Classification of zero sets of holomorphic functions	p. 106
4.4 Singularities of a function	p. 108
4.4.a Classification of singularities	p. 108
4.4.b Meromorphic functions	p. 110
4.5 Laurent series	p. 111
4.5.a Introduction and definition	p. 111
4.5.b Examples of Laurent series	p. 113
4.5.c The Residue theorem	p. 114
4.5.d Practical computations of residues	p. 116
4.6 Applications to the computation of horrifying integrals or ghastly sums	p. 117
4.6.a Jordan's lemmas	p. 117
4.6.b Integrals on R of a rational function	p. 118
4.6.c Fourier integrals	p. 120
4.6.d Integral on the unit circle of a rational function	p. 121
4.6.e Computation of infinite sums	p. 122
Exercises	p. 125
Problem	p. 128
Solutions	p. 129
5 Complex Analysis II	p. 135
5.1 Complex logarithm; multivalued functions	p. 135
5.1.a The complex logarithms	p. 135
5.1.b The square root function	p. 137
5.1.c Multivalued functions, Riemann surfaces	p. 137
5.2 Harmonic functions	p. 139
5.2.a Definitions	p. 139
5.2.b Properties	p. 140
5.2.c A trick to find f knowing u	p. 142
5.3 Analytic continuation	p. 144
5.4 Singularities at infinity	p. 146
5.5 The saddle point method	p. 148
5.5.a The general saddle point method	p. 149
5.5.b The real saddle point method	p. 152
Exercises	p. 153
Solutions	p. 154
6 Conformal maps	p. 155
6.1 Conformal maps	p. 155
6.1.a Preliminaries	p. 155
6.1.b The Riemann mapping theorem	p. 157
6.1.c Examples of conformal maps	p. 158
6.1.d The Schwarz-Christoffel transformation	p. 161
6.2 Applications to potential theory	p. 163
6.2.a Application to electrostatics	p. 165
6.2.b Application to hydrodynamics	p. 167
6.2.c Potential theory, lightning rods, and percolation	p. 169
6.3 Dirichlet problem and Poisson kernel	p. 170
Exercises	p. 174
Solutions	p. 176
7 Distributions I	p. 179
7.1 Physical approach	p. 179
7.1.a The problem of distribution of charge	p. 179
7.1.b The problem of momentum and forces during an elastic shock	p. 181
7.2 Definitions and examples of distributions	p. 182
7.2.a Regular distributions	p. 184
7.2.b Singular distributions	p. 185
7.2.c Support of a distribution	p. 187
7.2.d Other examples	p. 187
7.3 Elementary properties. Operations	p. 188
7.3.a Operations on distributions	p. 188
7.3.b Derivative of a distribution	p. 191
7.4 Dirac and its derivatives	p. 193
7.4.a The Heaviside distribution	p. 193
7.4.b Multidimensionai Dirac distributions	p. 194
7.4.c The distribution [delta]'	p. 196
7.4.d Composition of [delta] with a function	p. 198
7.4.e Charge and current densities	p. 199
7.5 Derivation of a discontinuous function	p. 201
7.5.a Derivation of a function discontinuous at a point	p. 201
7.5.b Derivative of a function with discontinuity along a surface L	p. 204
7.5.c Laplacian of a function discontinuous along a surface L	p. 206
7.5.d Application: laplacian of 1/r in 3-space	p. 207
7.6 Convolution	p. 209
7.6.a The tensor product of two functions	p. 209
7.6.b The tensor product of distributions	p. 209
7.6.c Convolution of two functions	p. 211
7.6.d "Fuzzy" measurement	p. 213
7.6.e Convolution of distributions	p. 214
7.6.f Applications	p. 215
7.6.g The Poisson equation	p. 216
7.7 Physical interpretation of convolution operators	p. 217
7.8 Discrete convolution	p. 220
8 Distributions II	p. 223
8.1 Cauchy principal value	p. 223
8.1.a Definition	p. 223
8.1.b Application to the computation of certain integrals	p. 224
8.1.c Feynman's notation	p. 225
8.1.d Kramers-Kronig relations	p. 227
8.1.e A few equations in the sense of distributions	p. 229
8.2 Topology D'	p. 230
8.2.a Weak convergence in D'	p. 230
8.2.b Sequences of functions converging to [delta]	p. 231
8.2.c Convergence in D' and convergence in the sense of functions	p. 234
8.2.d Regularization of a distribution	p. 234
8.2.e Continuity of convolution	p. 235
8.3 Convolution algebras	p. 236
8.4 Solving a differential equation with initial conditions	p. 238
8.4.a First order equations	p. 238
8.4.b The case of the harmonic oscillator	p. 239
8.4.c Other equations of physical origin	p. 240
Exercises	p. 241
Problem	p. 244
Solutions	p. 245
9 Hilbert spaces; Fourier series	p. 249
9.1 Insufficiency of vector spaces	p. 249
9.2 Pre-Hilbert spaces	p. 251
9.2.a The finite-dimensional case	p. 254
9.2.b Projection on a finite-dimensional subspace	p. 254
9.2.c Bessel inequality	p. 256
9.3 Hilbert spaces	p. 256
9.3.a Hilbert basis	p. 257
9.3.b The [ell superscript 2] space	p. 261
9.3.c The space L[superscript 2] [0,a]	p. 262
9.3.d The L[superscript 2](R) space	p. 263
9.4 Fourier series expansion	p. 264
9.4.a Fourier coefficients of a function	p. 264
9.4.b Mean-square convergence	p. 265
9.4.c Fourier series of a function f [Element] L[superscript 1] [0,a]	p. 266
9.4.d Pointwise convergence of the Fourier series	p. 267
9.4.e Uniform convergence of the Fourier series	p. 269
9.4.f The Gibbs phenomenon	p. 270
Exercises	p. 270
Problem	p. 271
Solutions	p. 272
10 Fourier transform of functions	p. 277
10.1 Fourier transform of a function in L[superscript 1]	p. 277
10.1.a Definition	p. 278
10.1.b Examples	p. 279
10.1.c The L[superscript 1] space	p. 279
10.1.d Elementary properties	p. 280
10.1.e Inversion	p. 282
10.1.f Extension of the inversion formula	p. 284
10.2 Properties of the Fourier transform	p. 285
10.2.a Transpose and translates	p. 285
10.2.b Dilation	p. 286
10.2.c Derivation	p. 286
10.2.d Rapidly decaying functions	p. 288
10.3 Fourier transform of a function in L[superscript 2]	p. 288
10.3.a The space L	p. 289
10.3.b The Fourier transform in L[superscript 2]	p. 290
10.4 Fourier transform and convolution	p. 292
10.4.a Convolution formula	p. 292
10.4.b Cases of the convolution formula	p. 293
Exercises	p. 295
Solutions	p. 296
11 Fourier transform of distributions	p. 299
11.1 Definition and properties	p. 299
11.1.a Tempered distributions	p. 300
11.1.b Fourier transform of tempered distributions	p. 301
11.1.c Examples	p. 303
11.1.d Higher-dimensional Fourier transforms	p. 305
11.1.e Inversion formula	p. 306
11.2 The Dirac comb	p. 307
11.2.a Definition and properties	p. 307
11.2.b Fourier transform of a periodic function	p. 308
11.2.c Poisson summation formula	p. 309
11.2.d Application to the computation of series	p. 310
11.3 The Gibbs phenomenon	p. 311
11.4 Application to physical optics	p. 314
11.4.a Link between diaphragm and diffraction figure	p. 314
11.4.b Diaphragm made of infinitely many infinitely narrow slits	p. 315
11.4.c Finite number of infinitely narrow slits	p. 316
11.4.d Finitely many slits with finite width	p. 318
11.4.e Circular lens	p. 320
11.5 Limitations of Fourier analysis and wavelets	p. 321
Exercises	p. 324
Problem	p. 325
Solutions	p. 326
12 The Laplace transform	p. 331
12.1 Definition and integrability	p. 331
12.1.a Definition	p. 332
12.1.b Integrability	p. 333
12.1.c Properties of the Laplace transform	p. 336
12.2 Inversion	p. 336
12.3 Elementary properties and examples of Laplace transforms	p. 338
12.3.a Translation	p. 338
12.3.b Convolution	p. 339
12.3.c Differentiation and integration	p. 339
12.3.d Examples	p. 341
12.4 Laplace transform of distributions	p. 342
12.4.a Definition	p. 342
12.4.b Properties	p. 342
12.4.c Examples	p. 344
12.4.d The z-transform	p. 344
12.4.e Relation between Laplace and Fourier transforms	p. 345
12.5 Physical applications, the Cauchy problem	p. 346
12.5.a Importance of the Cauchy problem	p. 346
12.5.b A simple example	p. 347
12.5.c Dynamics of the electromagnetic field without sources	p. 348
Exercises	p. 351
Solutions	p. 352
13 Physical applications of the Fourier transform	p. 355
13.1 Justification of sinusoidal regime analysis	p. 355
13.2 Fourier transform of vector fields: longitudinal and transverse fields	p. 358
13.3 Heisenberg uncertainty relations	p. 359
13.4 Analytic signals	p. 365
13.5 Autocorrelation of a finite energy function	p. 368
13.5.a Definition	p. 368
13.5.b Properties	p. 368
13.5.c Intercorrelation	p. 369
13.6 Finite power functions	p. 370
13.6.a Definitions	p. 370
13.6.b Autocorrelation	p. 370
13.7 Application to optics: the Wiener-Khintchine theorem	p. 371
Exercises	p. 375
Solutions	p. 376
14 Bras, kets, and all that sort of thing	p. 377
14.1 Reminders about finite dimension	p. 377
14.1.a Scalar product and representation theorem	p. 377
14.1.b Adjoint	p. 378
14.1.c Symmetric and hermitian endomorphisms	p. 379
14.2 Kets and bras	p. 379
14.2.a Kets [Characters not reproducible] [Element] H	p. 379
14.2.b Bras [Characters not reproducible] [Element] H'	p. 380
14.2.c Generalized bras	p. 382
14.2.d Generalized kets	p. 383
14.2.e Id = [Sigma subscript n] \| [phi subscript n]>	p. 384
14.2.f Generalized basis	p. 385
14.3 Linear operators	p. 387
14.3.a Operators	p. 387
14.3.b Adjoint	p. 389
14.3.c Bounded operators, closed operators, closable operators	p. 390
14.3.d Discrete and continuous spectra	p. 391
14.4 Hermitian operators; self-adjoint operators	p. 393
14.4.a Definitions	p. 394
14.4.b Eigenvectors	p. 396
14.4.c Generalized eigenvectors	p. 397
14.4.d "Matrix" representation	p. 398
14.4.e Summary of properties of the operators P and X	p. 401
Exercises	p. 403
Solutions	p. 404
15 Green functions	p. 407
15.1 Generalities about Green functions	p. 407
15.2 A pedagogical example: the harmonic oscillator	p. 409
15.2.a Using the Laplace transform	p. 410
15.2.b Using the Fourier transform	p. 410
15.3 Electromagnetism and the d'Alembertian operator	p. 414
15.3.a Computation of the advanced and retarded Green functions	p. 414
15.3.b Retarded potentials	p. 418
15.3.c Covariant expression of advanced and retarded Green functions	p. 421
15.3.d Radiation	p. 421
15.4 The heat equation	p. 422
15.4.a One-dimensional case	p. 423
15.4.b Three-dimensional case	p. 426
15.5 Quantum mechanics	p. 427
15.6 Klein-Gordon equation	p. 429
Exercises	p. 432
16 Tensors	p. 433
16.1 Tensors in affine space	p. 433
16.1.a Vectors	p. 433
16.1.b Einstein convention	p. 435
16.1.c Linear forms	p. 436
16.1.d Linear maps	p. 438
16.1.e Lorentz transformations	p. 439
16.2 Tensor product of vector spaces: tensors	p. 439
16.2.a Existence of the tensor product of two vector spaces	p. 439
16.2.b Tensor product of linear forms: tensors of type [Characters not reproducible]	p. 441
16.2.c Tensor product of vectors: tensors of type [Characters not reproducible]	p. 443
16.2.d Tensor product of a vector and a linear form: linear maps or [Characters not reproducible]-tensors	p. 444
16.2.e Tensors of type [Characters not reproducible]	p. 446
16.3 The metric, or, how to raise and lower indices	p. 447
16.3.a Metric and pseudo-metric	p. 447
16.3.b Natural duality by means of the metric	p. 449
16.3.c Gymnastics: raising and lowering indices	p. 450
16.4 Operations on tensors	p. 453
16.5 Change of coordinates	p. 455
16.5.a Curvilinear coordinates	p. 455
16.5.b Basis vectors	p. 456
16.5.c Transformation of physical quantities	p. 458
16.5.d Transformation of linear forms	p. 459
16.5.e Transformation of an arbitrary tensor field	p. 460
16.5.f Conclusion	p. 461
Solutions	p. 462
17 Differential forms	p. 463
17.1 Exterior algebra	p. 463
17.1.a 1-forms	p. 463
17.1.b Exterior 2-forms	p. 464
17.1.c Exterior k-forms	p. 465
17.1.d Exterior product	p. 467
17.2 Differential forms on a vector space	p. 469
17.2.a Definition	p. 469
17.2.b Exterior derivative	p. 470
17.3 Integration of differential forms	p. 471
17.4 Poincare's theorem	p. 474
17.5 Relations with vector calculus: gradient, divergence, curl	p. 476
17.5.a Differential forms in dimension 3	p. 476
17.5.b Existence of the scalar electrostatic potential	p. 477
17.5.c Existence of the vector potential	p. 479
17.5.d Magnetic monopoles	p. 480
17.6 Electromagnetism in the language of differential forms	p. 480
Problem	p. 484
Solution	p. 485
18 Groups and group representations	p. 489
18.1 Groups	p. 489
18.2 Linear representations of groups	p. 491
18.3 Vectors and the group SO(3)	p. 492
18.4 The group SU(2) and spinors	p. 497
18.5 Spin and Riemann sphere	p. 503
Exercises	p. 505
19 Introduction to probability theory	p. 509
19.1 Introduction	p. 510
19.2 Basic definitions	p. 512
19.3 Poincare formula	p. 516
19.4 Conditional probability	p. 517
19.5 Independent events	p. 519
20 Random variables	p. 521
20.1 Random variables and probability distributions	p. 521
20.2 Distribution function and probability density	p. 524
20.2.a Discrete random variables	p. 526
20.2.b (Absolutely) continuous random variables	p. 526
20.3 Expectation and variance	p. 527
20.3.a Case of a discrete r.v.	p. 527
20.3.b Case of a continuous r.v.	p. 528
20.4 An example: the Poisson distribution	p. 530
20.4.a Particles in a confined gas	p. 530
20.4.b Radioactive decay	p. 531
20.5 Moments of a random variable	p. 532
20.6 Random vectors	p. 534
20.6.a Pair of random variables	p. 534
20.6.b Independent random variables	p. 537
20.6.c Random vectors	p. 538
20.7 Image measures	p. 539
20.7.a Case of a single random variable	p. 539
20.7.b Case of a random vector	p. 540
20.8 Expectation and characteristic function	p. 540
20.8.a Expectation of a function of random variables	p. 540
20.8.b Moments, variance	p. 541
20.8.c Characteristic function	p. 541
20.8.d Generating function	p. 543
20.9 Sum and product of random variables	p. 543
20.9.a Sum of random variables	p. 543
20.9.b Product of random variables	p. 546
20.9.c Example: Poisson distribution	p. 547
20.10 Bienayme-Tchebychev inequality	p. 547\
20.10.a Statement	p. 547
20.10.b Application: Buffon's needle	p. 549
20.11 Independance, correlation, causality	p. 550
21 Convergence of random variables: central limit theorem	p. 553
21.1 Various types of convergence	p. 553
21.2 The law of large numbers	p. 555
21.3 Central limit theorem	p. 556
Exercises	p. 560
Problems	p. 563
Solutions	p. 564
Appendices
A Reminders concerning topology and normed vector spaces	p. 573
A.1 Topology, topological spaces	p. 573
A.2 Normed vector spaces	p. 577
A.2.a Norms, seminorms	p. 577
A.2.b Balls and topology associated to the distance	p. 578
A.2.c Comparison of sequences	p. 580
A.2.d Bolzano-Weierstrass theorems	p. 581
A.2.e Comparison of norms	p. 581
A.2.f Norm of a linear map	p. 583
Exercise	p. 583
Solution	p. 584
B Elementary reminders of differential calculus	p. 585
B.1 Differential of a real-valued function	p. 585
B.1.a Functions of one real variable	p. 585
B.1.b Differential of a function f : R[superscript n] [right arrow] R	p. 586
B.1.c Tensor notation	p. 587
B.2 Differential of map with values in R[superscript p]	p. 587
B.3 Lagrange multipliers	p. 588
Solution	p. 591
C Matrices	p. 593
C.1 Duality	p. 593
C.2 Application to matrix representation	p. 594
C.2.a Matrix representing a family of vectors	p. 594
C.2.b Matrix of a linear map	p. 594
C.2.c Change of basis	p. 595
C.2.d Change of basis formula	p. 595
C.2.e Case of an orthonormal basis	p. 596
D A few proofs	p. 597
Tables
Fourier transforms	p. 609
Laplace transforms	p. 613
Probability laws	p. 616
Further reading	p. 617
References	p. 621
Portraits	p. 627
Sidebars	p. 629
Index	p. 631

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