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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