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ОбложкаRiley K.F. Foundation mathematics for the physical sciences / K.F.Riley, M.P.Hobson. - Cambridge; New York: Cambridge University Press, 2011. - xiii, 721 p.: ill. - Ind.: p.706-721. - ISBN 978-0-521-19273-6
Шифр: (И/В1-R57) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
 
   Preface ..................................................... xi

1  Arithmetic and geometry ...................................... 1
   1.1  Powers .................................................. 1
   1.2  Exponential and logarithmic functions ................... 7
   1.3  Physical dimensions .................................... 15
   1.4  The binomial expansion ................................. 20
   1.5  Trigonometric identities ............................... 24
   1.6  Inequalities ........................................... 32
   Summary ..................................................... 40
   Problems .................................................... 42
   Hints and answers ........................................... 49

2  Preliminary algebra ......................................... 52
   2.1  Polynomials and polynomial equations ................... 53
   2.2  Coordinate geometry .................................... 64
   2.3  Partial fractions ...................................... 74
   2.4  Some particular methods of proof ....................... 84
   Summary ..................................................... 91
   Problems .................................................... 93
   Hints and answers ........................................... 99

3  Differential calculus ...................................... 102
   3.1  Differentiation ....................................... 102
   3.2  Leibnitz's theorem .................................... 112
   3.3  Special points of a function .......................... 114
   3.4  Curvature of a function ............................... 116
   3.5  Theorems of differentiation ........................... 120
   3.6  Graphs ................................................ 124
   Summary .................................................... 133
   Problems ................................................... 134
   Hints and answers .......................................... 138

4  Integral calculus .......................................... 141
   4.1  Integration ........................................... 141
   4.2  Integration methods ................................... 146
   4.3  Integration by parts .................................. 152
   4.4  Reduction formulae .................................... 155
   4.5  Infinite and improper integrals ....................... 156
   4.6  Integration in plane polar coordinates ................ 159
   4.7  Integral inequalities ................................. 160
   4.8  Applications of integration ........................... 161
   Summary .................................................... 168
   Problems ................................................... 170
   Hints and answers .......................................... 173

5  Complex numbers and hyperbolic functions ................... 174
   5.1  The need for complex numbers .......................... 174
   5.2  Manipulation of complex numbers ....................... 176
   5.3  Polar representation of complex numbers ............... 185
   5.4  De Moivre's theorem ................................... 189
   5.5  Complex logarithms and complex powers ................. 194
   5.6  Applications to differentiation and integration ....... 196
   5.7  Hyperbolic functions .................................. 197
   Summary .................................................... 205
   Problems ................................................... 206
   Hints and answers .......................................... 211

6  Series and limits .......................................... 213
   6.1  Series ................................................ 213
   6.2  Summation of series ................................... 215
   6.3  Convergence of infinite series ........................ 224
   6.4  Operations with series ................................ 232
   6.5  Power series .......................................... 233
   6.6  Taylor series ......................................... 238
   6.7  Evaluation of limits .................................. 244
   Summary .................................................... 248
   Problems ................................................... 250
   Hints and answers .......................................... 257

7  Partial differentiation .................................... 259
   7.1  Definition of the partial derivative .................. 259
   7.2  The total differential and total derivative ........... 261
   7.3  Exact and inexact differentials ....................... 264
   7.4  Useful theorems of partial differentiation ............ 266
   7.5  The chain rule ........................................ 267
   7.6  Change of variables ................................... 268
   7.7  Taylor's theorem for many-variable functions .......... 270
   7.8  Stationary values of two-variable functions ........... 272
   7.9  Stationary values under constraints ................... 276
   7.10 Envelopes ............................................. 282
   7.11 Thermodynamic relations ............................... 285
   7.12 Differentiation of integrals .......................... 288
   Summary .................................................... 290
   Problems ................................................... 292
   Hints and answers .......................................... 299

8  Multiple integrals ......................................... 301
   8.1  Double integrals ...................................... 301
   8.2  Applications of multiple integrals .................... 305
   8.3  Change of variables in multiple integrals ............. 315
   Summary .................................................... 324
   Problems ................................................... 325
   Hints and answers .......................................... 329

9  Vector algebra ............................................. 331
   9.1  Scalars and vectors ................................... 331
   9.2  Addition, subtraction and multiplication of vectors ... 332
   9.3  Basis vectors, components and magnitudes .............. 336
   9.4  Multiplication of two vectors ......................... 339
   9.5  Triple products ....................................... 346
   9.6  Equations of lines, planes and spheres ................ 348
   9.7  Using vectors to find distances ....................... 353
   9.8  Reciprocal vectors .................................... 357
   Summary .................................................... 359
   Problems ................................................... 361
   Hints and answers .......................................... 368

10 Matrices and vector spaces ................................. 369
   10.1 Vector spaces ......................................... 370
   10.2 Linear operators ...................................... 374
   10.3 Matrices .............................................. 376
   10.4 Basic matrix algebra .................................. 377
   10.5 The transpose and conjugates of a matrix .............. 383
   10.6 The trace of a matrix ................................. 385
   10.7 The determinant of a matrix ........................... 386
   10.8 The inverse of a matrix ............................... 392
   10.9 The rank of a matrix .................................. 395
   10.10 Simultaneous linear equations ........................ 397
   10.11 Special types of square matrix ....................... 408
   10.12 Eigenvectors and eigenvalues ......................... 412
   10.13 Determination of eigenvalues and eigenvectors ........ 418
   10.14 Change of basis and similarity transformations ....... 421
   10.15 Diagonalisation of matrices .......................... 424
   10.16 Quadratic and Hermitian forms ........................ 427
   10.17 The summation convention ............................. 432
   Summary .................................................... 433
   Problems ................................................... 437
   Hints and answers .......................................... 445

11 Vector calculus ............................................ 448
   11.1 Differentiation of vectors ............................ 448
   11.2 Integration of vectors ................................ 453
   11.3 Vector functions of several arguments ................. 454
   11.4 Surfaces .............................................. 455
   11.5 Scalar and vector fields .............................. 458
   11.6 Vector operators ...................................... 458
   11.7 Vector operator formulae .............................. 465
   11.8 Cylindrical and spherical polar coordinates ........... 469
   11.9 General curvilinear coordinates ....................... 476
   Summary .................................................... 482
   Problems ................................................... 483
   Hints and answers .......................................... 490

12 Line, surface and volume integrals ......................... 491
   12.1 Line integrals ........................................ 491
   12.2 Connectivity of regions ............................... 497
   12.3 Green's theorem in a plane ............................ 498
   12.4 Conservative fields and potentials .................... 502
   12.5 Surface integrals ..................................... 504
   12.6 Volume integrals ...................................... 511
   12.7 Integral forms for grad, div and curl ................. 513
   12.8 Divergence theorem and related theorems ............... 517
   12.9 Stokes'theorem and related theorems ................... 523
   Summary .................................................... 527
   Problems ................................................... 528
   Hints and answers .......................................... 534

13 Laplace transforms ......................................... 536
   13.1 Laplace transforms .................................... 537
   13.2 The Dirac 5-function and Heaviside step function ...... 541
   13.3 Laplace transforms of derivatives and integrals ....... 544
   13.4 Other properties of Laplace transforms ................ 546
   Summary .................................................... 549
   Problems ................................................... 550
   Hints and answers .......................................... 552

14 Ordinary differential equations ............................ 554
   14.1 General form of solution .............................. 555
   14.2 First-degree first-order equations .................... 557
   14.3 Higher degree first-order equations ................... 565
   14.4 Higher order linear ODEs .............................. 569
   14.5 Linear equations with constant coefficients ........... 572
   14.6 Linear recurrence relations ........................... 579
   Summary .................................................... 585
   Problems ................................................... 587
   Hints and answers .......................................... 595

15 Elementary probability ..................................... 597
   15.1 Venn diagrams ......................................... 597
   15.2 Probability ........................................... 602
   15.3 Permutations and combinations ......................... 612
   15.4 Random variables and distributions .................... 618
   15.5 Properties of distributions ........................... 623
   15.6 Functions of random variables ......................... 628
   15.7 Important discrete distributions ...................... 632
   15.8 Important continuous distributions .................... 643
   15.9 Joint distributions ................................... 655
   Summary .................................................... 661
   Problems ................................................... 664
   Hints and answers .......................................... 670

A  The base for natural logarithms ............................ 673
В  Sinusoidal definitions ..................................... 676
С  Leibnitz's theorem ......................................... 679
D  Summation convention ....................................... 681
E  Physical constants ......................................... 684
F  Footnote answers ........................................... 685

   Index ...................................................... 706

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