Foundation Mathematics for the Physical Sciences

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 13
1.1 Powers......Page 17
1.2 Exponential and logarithmic functions......Page 23
1.2.1 Logarithms......Page 24
1.2.2 The exponential function and choice of logarithmic base......Page 25
1.2.3 The use of logarithms......Page 27
1.3 Physical dimensions......Page 31
1.4 The binomial expansion......Page 36
1.4.1 Binomial coefficients......Page 37
1.4.3 Negative and non-integral values of n......Page 38
1.4.4 Relationship with the exponential function......Page 39
1.5 Trigonometric identities......Page 40
1.5.1 Compound-angle identities......Page 44
1.5.2 Double- and half-angle identities......Page 46
1.6 Inequalities......Page 48
Summary......Page 56
Problems......Page 58
Hints and answers......Page 65
2 Preliminary algebra......Page 68
2.1 Polynomials and polynomial equations......Page 69
2.1.1 Example: the cubic case......Page 71
2.1.2 A more general case......Page 75
2.1.3 Factorising polynomials......Page 76
2.1.4 Properties of roots......Page 78
2.2.1 Linear graphs......Page 80
2.2.2 Conic sections......Page 81
2.2.3 Parametric equations......Page 85
2.2.4 Plane polar coordinates......Page 88
2.3 Partial fractions......Page 90
2.3.1 The general method......Page 92
2.3.2 Complications and special cases......Page 95
2.4 Some particular methods of proof......Page 100
2.4.1 Proof by induction......Page 101
2.4.2 Proof by contradiction......Page 103
2.4.3 Necessary and sufficient conditions......Page 104
Summary......Page 107
Problems......Page 109
Hints and answers......Page 115
3.1.1 Differentiation from first principles......Page 118
3.1.2 Differentiation of products......Page 122
3.1.3 The chain rule......Page 124
3.1.4 Differentiation of quotients......Page 125
3.1.5 Implicit differentiation......Page 126
3.1.6 Logarithmic differentiation......Page 127
3.2 Leibnitz\'s theorem......Page 128
3.3 Special points of a function......Page 130
3.4 Curvature of a function......Page 132
3.5.2 Mean value theorem......Page 136
3.5.3 Applications of Rolle\'s theorem and the mean value theorem......Page 137
3.6 Graphs......Page 140
3.6.1 General considerations......Page 141
3.6.2 Worked examples......Page 143
Summary......Page 149
Problems......Page 150
Hints and answers......Page 154
4.1 Integration......Page 157
4.1.1 Integration from first principles......Page 158
4.1.2 Integration as the inverse of differentiation......Page 159
4.2.2 Integration of sinusoidal functions......Page 162
4.2.4 Integration using partial fractions......Page 164
4.2.5 Integration by substitution......Page 165
4.3 Integration by parts......Page 168
4.4 Reduction formulae......Page 171
4.5 Infinite and improper integrals......Page 172
4.6 Integration in plane polar coordinates......Page 175
4.7 Integral inequalities......Page 176
4.8.1 Mean value of a function......Page 177
4.8.2 Finding the length of a curve......Page 178
4.8.3 Surfaces of revolution......Page 180
4.8.4 Volumes of revolution......Page 182
Summary......Page 184
Problems......Page 186
Hints and answers......Page 189
5.1 The need for complex numbers......Page 190
5.2 Manipulation of complex numbers......Page 192
5.2.1 Addition and subtraction......Page 193
5.2.2 Modulus and argument......Page 194
5.2.3 Multiplication......Page 195
5.2.4 Complex conjugate......Page 197
5.2.5 Division......Page 200
5.3 Polar representation of complex numbers......Page 201
5.3.1 Multiplication and division in polar form......Page 203
5.4.1 Trigonometric identities......Page 205
5.4.2 Finding the nth roots of unity......Page 207
5.4.3 Solving polynomial equations......Page 208
5.5 Complex logarithms and complex powers......Page 210
5.6 Applications to differentiation and integration......Page 212
5.7 Hyperbolic functions......Page 213
5.7.1 Definitions......Page 214
5.7.2 Hyperbolic--trigonometric analogies......Page 215
5.7.4 Solving hyperbolic equations......Page 216
5.7.5 Inverses of hyperbolic functions......Page 217
5.7.6 Calculus of hyperbolic functions......Page 218
Summary......Page 221
Problems......Page 222
Hints and answers......Page 227
6.1 Series......Page 229
6.2.2 Geometric series......Page 231
6.2.4 The difference method......Page 233
6.2.5 Series involving natural numbers......Page 236
6.2.6 Transformation of series......Page 237
6.3 Convergence of infinite series......Page 240
6.3.1 Absolute and conditional convergence......Page 241
6.3.2 Convergence of a series containing only real positive terms......Page 242
6.3.3 Alternating series test......Page 247
6.4 Operations with series......Page 248
6.5 Power series......Page 249
6.5.1 Convergence of power series......Page 250
6.5.2 Operations with power series......Page 252
6.6 Taylor series......Page 254
6.6.1 Taylor\'s theorem......Page 255
6.6.2 Approximation errors in Taylor series......Page 258
6.6.3 Standard Maclaurin series......Page 259
6.7 Evaluation of limits......Page 260
Summary......Page 264
Problems......Page 266
Hints and answers......Page 273
7.1 Definition of the partial derivative......Page 275
7.2 The total differential and total derivative......Page 277
7.3 Exact and inexact differentials......Page 280
7.4 Useful theorems of partial differentiation......Page 282
7.5 The chain rule......Page 283
7.6 Change of variables......Page 284
7.7 Taylor\'s theorem for many-variable functions......Page 286
7.8 Stationary values of two-variable functions......Page 288
7.9 Stationary values under constraints......Page 292
7.10 Envelopes......Page 298
7.11 Thermodynamic relations......Page 301
7.12 Differentiation of integrals......Page 304
Summary......Page 306
Problems......Page 308
Hints and answers......Page 315
8.1 Double integrals......Page 317
8.2 Applications of multiple integrals......Page 321
8.2.1 Areas and volumes......Page 322
8.2.2 Masses, centres of mass and centroids......Page 324
8.2.3 Pappus\'s theorems......Page 326
8.2.4 Moments of inertia......Page 328
8.2.5 Mean values of functions......Page 329
8.3.1 Change of variables in double integrals......Page 331
8.3.2 Evaluation of the integral…......Page 334
8.3.3 Change of variables in triple integrals......Page 336
8.3.4 General properties of Jacobians......Page 338
Summary......Page 340
Problems......Page 341
Hints and answers......Page 345
9.1 Scalars and vectors......Page 347
9.2 Addition, subtraction and multiplication of vectors......Page 348
9.3 Basis vectors, components and magnitudes......Page 352
9.4.1 Scalar product......Page 355
9.4.2 Vector product......Page 358
9.5.1 Scalar triple product......Page 362
9.6 Equations of lines, planes and spheres......Page 364
9.6.2 Equation of a plane......Page 365
9.6.3 Equation of a sphere......Page 367
9.7.1 Distance from a point to a line......Page 369
9.7.2 Distance from a point to a plane......Page 370
9.7.3 Distance from a line to a line......Page 371
9.7.4 Distance from a line to a plane......Page 372
9.8 Reciprocal vectors......Page 373
Summary......Page 375
Problems......Page 377
Hints and answers......Page 384
10 Matrices and vector spaces......Page 385
10.1.1 Basis vectors......Page 386
10.1.2 The inner product......Page 387
10.1.3 Some useful inequalities......Page 389
10.2 Linear operators......Page 390
10.3 Matrices......Page 392
10.4 Basic matrix algebra......Page 393
10.4.1 Matrix addition and multiplication by a scalar......Page 394
10.4.2 Multiplication of matrices......Page 395
10.4.3 The null and identity matrices......Page 397
10.4.4 Functions of matrices......Page 398
10.5 The transpose and conjugates of a matrix......Page 399
10.5.1 The complex and Hermitian conjugates......Page 400
10.6 The trace of a matrix......Page 401
10.7 The determinant of a matrix......Page 402
10.7.1 Properties of determinants......Page 405
10.7.2 Evaluation of determinants......Page 406
10.8 The inverse of a matrix......Page 408
10.9 The rank of a matrix......Page 411
10.10 Simultaneous linear equations......Page 413
10.10.1 The number of solutions......Page 414
10.10.2 N simultaneous linear equations in N unknowns......Page 415
10.10.3 A geometrical interpretation......Page 422
10.11.2 Lower and upper triangular matrices......Page 424
10.11.3 Symmetric and antisymmetric matrices......Page 425
10.11.6 Unitary matrices......Page 426
10.11.7 Normal matrices......Page 427
10.12 Eigenvectors and eigenvalues......Page 428
10.12.1 Eigenvectors and eigenvalues of Hermitian and unitary matrices......Page 430
10.12.2 Eigenvectors and eigenvalues of a general square matrix......Page 431
10.12.3 Simultaneous eigenvectors......Page 432
10.13 Determination of eigenvalues and eigenvectors......Page 434
10.14 Change of basis and similarity transformations......Page 437
10.15 Diagonalisation of matrices......Page 440
10.16 Quadratic and Hermitian forms......Page 443
10.16.1 The stationary properties of the eigenvectors......Page 445
10.16.2 Quadratic surfaces......Page 446
10.17 The summation convention......Page 448
Summary......Page 449
Problems......Page 453
Hints and answers......Page 461
11.1 Differentiation of vectors......Page 464
11.1.1 Differentiation of composite vector expressions......Page 467
11.1.2 Differential of a vector......Page 468
11.2 Integration of vectors......Page 469
11.3 Vector functions of several arguments......Page 470
11.4 Surfaces......Page 471
11.6 Vector operators......Page 474
11.6.1 Gradient of a scalar field......Page 475
11.6.2 Divergence of a vector field......Page 478
11.6.3 Curl of a vector field......Page 479
11.7 Vector operator formulae......Page 481
11.7.1 Vector operators acting on sums and products......Page 482
11.7.2 Combinations of grad, div and curl......Page 483
11.8 Cylindrical and spherical polar coordinates......Page 485
11.8.1 Cylindrical polar coordinates......Page 486
11.8.2 Spherical polar coordinates......Page 489
11.9 General curvilinear coordinates......Page 492
Summary......Page 498
Problems......Page 499
Hints and answers......Page 506
12.1 Line integrals......Page 507
12.1.1 Evaluating line integrals......Page 508
12.1.2 Physical examples of line integrals......Page 511
12.1.3 Line integrals with respect to a scalar......Page 512
12.2 Connectivity of regions......Page 513
12.3 Green\'s theorem in a plane......Page 514
12.4 Conservative fields and potentials......Page 518
12.5 Surface integrals......Page 520
12.5.1 Evaluating surface integrals......Page 522
12.5.2 Vector areas of surfaces......Page 524
12.5.3 Physical examples of surface integrals......Page 526
12.6 Volume integrals......Page 527
12.7 Integral forms for grad, div and curl......Page 529
12.8 Divergence theorem and related theorems......Page 533
12.8.1 Green\'s theorems......Page 534
12.8.2 Other related integral theorems......Page 535
12.8.3 Physical applications of the divergence theorem......Page 536
12.9 Stokes\' theorem and related theorems......Page 539
12.9.1 Related integral theorems......Page 540
12.9.2 Physical applications of Stokes\' theorem......Page 541
Summary......Page 543
Problems......Page 544
Hints and answers......Page 550
13 Laplace transforms......Page 552
13.1 Laplace transforms......Page 553
13.2 The Dirac delta-function and Heaviside step function......Page 557
13.3 Laplace transforms of derivatives and integrals......Page 560
13.4 Other properties of Laplace transforms......Page 562
Summary......Page 565
Problems......Page 566
Hints and answers......Page 568
14 Ordinary differential equations......Page 570
14.1 General form of solution......Page 571
14.2.1 Separable-variable equations......Page 573
14.2.2 Exact equations......Page 574
14.2.3 Inexact equations: integrating factors......Page 575
14.2.4 Linear equations......Page 577
14.2.5 Homogeneous equations......Page 578
14.2.6 Bernoulli\'s equation......Page 579
14.3.1 Equations soluble for p......Page 581
14.3.2 Equations soluble for x......Page 582
14.3.3 Equations soluble for y......Page 583
14.4 Higher order linear ODEs......Page 585
14.5.1 Finding the complementary function yc(x)......Page 588
14.5.2 Finding the particular integral yp(x)......Page 590
14.5.3 Constructing the general solution yc(x)+yp(x)......Page 591
14.5.4 Laplace transform method......Page 592
14.6 Linear recurrence relations......Page 595
14.6.1 First-order recurrence relations......Page 596
14.6.2 Second-order recurrence relations......Page 598
14.6.3 Higher order recurrence relations......Page 600
Summary......Page 601
Problems......Page 603
Hints and answers......Page 611
15.1 Venn diagrams......Page 613
15.2 Probability......Page 618
15.2.1 Axioms and theorems......Page 619
15.2.2 Conditional probability......Page 622
15.2.3 Bayes\' theorem......Page 626
15.3 Permutations and combinations......Page 628
15.3.1 Permutations......Page 629
15.3.2 Combinations......Page 630
15.4.1 Discrete random variables......Page 634
15.4.2 Continuous random variables......Page 636
15.4.3 Sets of random variables......Page 638
15.5 Properties of distributions......Page 639
15.5.1 Mean......Page 640
15.5.2 Mode and median......Page 641
15.5.4 Moments......Page 642
15.6 Functions of random variables......Page 644
15.6.1 Continuous random variables......Page 645
15.6.2 Expectation values and variances......Page 646
15.7.1 The binomial distribution......Page 648
15.7.2 The multinomial distribution......Page 651
15.7.3 The geometric and negative binomial distributions......Page 652
15.7.4 The hypergeometric distribution......Page 653
15.7.5 The Poisson distribution......Page 655
15.8 Important continuous distributions......Page 659
15.8.2 The Cauchy and Breit–Wigner distributions......Page 660
15.8.3 The Gaussian distribution......Page 661
15.8.5 The chi-squared distribution......Page 669
15.9.1 Bivariate distributions......Page 671
15.9.3 Means......Page 672
15.9.5 Covariance and correlation......Page 673
Summary......Page 677
Problems......Page 680
Hints and answers......Page 686
Appendix A: The base for natural logarithms......Page 689
Appendix B: Sinusoidal definitions......Page 692
Appendix C: Leibnitz’s theorem......Page 695
Appendix D: Summation convention......Page 697
Appendix E: Physical constants......Page 700
1. Arithmetic and geometry......Page 701
2. Preliminary algebra......Page 702
3. Differential calculus......Page 704
4. Integral calculus......Page 705
5. Complex numbers and hyperbolic functions......Page 706
6. Series and limits......Page 709
7. Partial differentiation......Page 711
8. Multiple integrals......Page 713
9. Vector algebra......Page 714
10. Matrices and vector spaces......Page 716
11. Vector calculus......Page 718
12. Line, surface and volume integrals......Page 719
14. Ordinary differential equations......Page 720
15. Elementary probability......Page 721
Index......Page 722