Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability

Preface ......Page 8
Contents ......Page 12
Part 1. Linear Analysis ......Page 26
1.2 The definition of a linear space ......Page 28
1.3 Examples of linear spaces ......Page 29
1.4 Elementary consequences of the axioms ......Page 31
1.5 Exercises ......Page 32
1.6 Subspaces of a linear space ......Page 33
1.7 Dependent and independent sets in a linear space ......Page 34
1.8 Bases and dimension ......Page 37
1.10 Exercises ......Page 38
1.11 Inner products, Euclidean spaces. Norms ......Page 39
1.12 Orthogonality in a Euclidean space ......Page 43
1.13 Exercises ......Page 45
1.14 Construction of orthogonal sets. The Gram-Schmidt process ......Page 47
1.15 Orthogonal complements. Projections ......Page 51
1.16 Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace ......Page 53
1.17 Exercises ......Page 55
2.1 Linear transformations ......Page 56
2.2 Null space and range ......Page 57
2.4 Exercises ......Page 60
2.5 Algebraic operations on linear transformations ......Page 61
2.6 Inverses ......Page 63
2.7 One-to-one linear transformations ......Page 66
2.8 Exercises ......Page 67
2.9 Linear transformations with prescribed values ......Page 69
2.10 Matrix representations of linear transformations ......Page 70
2.11 Construction of a matrix representation in diagonal form ......Page 73
2.12 Exercises ......Page 75
2.13 Linear spaces of matrices ......Page 76
2.14 Isomorphism between linear transformations and matrices ......Page 77
2.15 Multiplication of matrices ......Page 79
2.16 Exercises ......Page 82
2.17 Systems of linear equations ......Page 83
2.18 Computation techniques ......Page 86
2.19 Inverses of square matrices ......Page 90
2.20 Exercises ......Page 92
2.21 Miscellaneous exercises on matrices ......Page 93
3.1 Introduction ......Page 96
3.2 Motivation for the choice of axioms for a determinant function ......Page 97
3.3 A set of axioms for a determinant function ......Page 98
3.4 Computation of determinants ......Page 101
3.6 Exercises ......Page 104
3.7 The product formula for determinants ......Page 106
3.9 Determinants and independence of vectors ......Page 108
3.10 The determinant of a block-diagonal matrix ......Page 109
3.11 Exercises ......Page 110
3.12 Expansion formulas for determinants. Minors and cofactors ......Page 111
3.13 Existence of the determinant function ......Page 115
3.14 The determinant of a transpose ......Page 116
3.15 The cofactor matrix ......Page 117
3.16 Cramer's rule ......Page 118
3.17 Exercises ......Page 119
4.1 Linear transformations with diagonal matrix representations ......Page 121
4.2 Eigenvectors and eigenvalues of a linear transformation ......Page 122
4.3 Linear independence of eigenvectors corresponding to distinct eigenvalues ......Page 125
4.4 Exercises ......Page 126
4.5 The finite-dimensional case. Characteristic polynomials ......Page 127
4.6 Calculation of eigenvalues and eigenvectors in the finite-dimensional case ......Page 128
4.7 Trace of a matrix ......Page 131
4.8 Exercises ......Page 132
4.9 Matrices representing the same linear transformation. Similar matrices ......Page 133
4.10 Exercises ......Page 137
5.1 Eigenvalues and inner products ......Page 139
5.2 Hermitian and skew-Hermitian transformations ......Page 140
5.4 Orthogonality of eigenvectors corresponding to distinct eigenvalues ......Page 142
5.5 Exercises ......Page 143
5.6 Existence of an orthonormal set of eigenvectors for Hermitian and skew-Hermitian operators acting on finite-dimensional spaces ......Page 145
5.7 Matrix representations for Hermitian and skew-Hermitian operators ......Page 146
5.9 Diagonalization of a Hermitian or skew-Hermitian matrix ......Page 147
5.10 Unitary matrices. Orthogonal matrices ......Page 148
5.11 Exercises ......Page 149
5.12 Quadratic forms ......Page 151
5.13 Reduction of a real quadratic form to a diagonal form ......Page 153
5.14 Applications to analytic geometry ......Page 155
5.15 Exercises ......Page 159
* 5.16 Eigenvalues of a symmetric transformation obtained as values of its quadratic form ......Page 160
* 5.17 Extremal properties of eigenvalues of a symmetric transformation ......Page 161
* 5.18 The finite-dimensional case ......Page 162
5.19 Unitary transformations ......Page 163
5.20 Exercises ......Page 166
6.1 Historical introduction ......Page 167
6.2 Review of results concerning linear equations of first and second orders ......Page 168
6.3 Exercises ......Page 169
6.4 Linear differential equations of order n ......Page 170
6.6 The dimension of the solution space of a homogeneous linear equation ......Page 172
6.7 The algebra of constant-coefficient operators ......Page 173
6.8 Determination of a basis of solutions for linear equations with constant coefficients by factorization of operators ......Page 175
6.9 Exercises ......Page 179
6.10 The relation between the homogeneous and nonhomogeneous equations ......Page 181
6.11 Determination of a particular solution of the nonhomogeneous equation. The method of variation of parameters ......Page 182
6.12 Nonsingularity of the Wronskian matrix of n independent solutions of a homogeneous linear equation ......Page 186
6.14 The annihilator method for determining a particular solution of the nonhomogeneous equation ......Page 188
6.15 Exercises ......Page 191
6.16 Miscellaneous exercises on linear differential equations ......Page 192
6.17 Linear equations of second order with analytic coefficients ......Page 194
6.18 The Legendre equation ......Page 196
6.19 The Legendre polynomials ......Page 199
6.20 Rodrigues' formula for the Legendre polynomials ......Page 201
6.21 Exercises ......Page 202
6.22 The method of Frobenius ......Page 205
6.23 The Bessel equation ......Page 207
6.24 Exercises ......Page 213
7.1 Introduction ......Page 216
7.2 Calculus of matrix functions ......Page 218
7.3 Infinite series of matrices. Norms of matrices ......Page 219
7.4 Exercises ......Page 220
7.6 The differential equation satisfied by etA ......Page 222
7.7 Uniqueness theorem for the matrix differential equation F'{t) = AF(t) ......Page 223
7.8 The law of exponents for exponential matrices ......Page 224
7.9 Existence and uniqueness theorems for homogeneous linear systems with constant coefficients ......Page 225
7.10 The problem of calculating etA ......Page 226
7.11 The Cay ley-Hamilton theorem ......Page 228
7.13 Putzer's method for calculating etA ......Page 230
7.14 Alternate methods for calculating etA in special cases ......Page 233
7.15 Exercises ......Page 236
7.16 Nonhomogeneous linear systems with constant coefficients ......Page 238
7.17 Exercises ......Page 240
7.18 The general linear system Y\t) = P{t) Y(t) + Q(t) ......Page 242
7.19 A power-series method for solving homogeneous linear systems ......Page 245
7.20 Exercises ......Page 246
7.21 Proof of the existence theorem by the method of successive approximations ......Page 247
7.22 The method of successive approximations applied to first-order nonlinear systems ......Page 252
7.23 Proof of an existence-uniqueness theorem for first-order nonlinear systems ......Page 254
7.24 Exercises ......Page 255
* 7.25 Successive approximations and fixed points of operators ......Page 257
* 7.26 Normed linear spaces ......Page 258
* 7.27 Contraction operators ......Page 259
* 7.28 Fixed-point theorem for contraction operators ......Page 260
* 7.29 Applications of the fixed-point theorem ......Page 262
Part 2. Nonlinear Analysis ......Page 266
8.1 Functions from Rn to Rm. Scalar and vector fields ......Page 268
8.2 Open balls and open sets ......Page 269
8.3 Exercises ......Page 270
8.4 Limits and continuity ......Page 272
8.5 Exercises ......Page 276
8.6 The derivative of a scalar field with respect to a vector ......Page 277
8.7 Directional derivatives and partial derivatives ......Page 279
8.9 Exercises ......Page 280
8.10 Directional derivatives and continuity ......Page 282
8.11 The total derivative ......Page 283
8.12 The gradient of a scalar field ......Page 284
8.13 A sufficient condition for differentiability ......Page 286
8.14 Exercises ......Page 287
8.15 A chain rule for derivatives of scalar fields ......Page 288
8.16 Applications to geometry. Level sets. Tangent planes ......Page 291
8.17 Exercises ......Page 293
8.18 Derivatives of vector fields ......Page 294
8.19 Differentiability implies continuity ......Page 296
8.20 The chain rule for derivatives of vector fields ......Page 297
8.21 Matrix form of the chain rule ......Page 298
8.22 Exercises ......Page 300
* 8.23 Sufficient conditions for the equality of mixed partial derivatives ......Page 302
8.24 Miscellaneous exercises ......Page 306
9.1 Partial differential equations ......Page 308
9.2 A first-order partial differential equation with constant coefficients ......Page 309
9.3 Exercises ......Page 311
9.4 The one-dimensional wave equation ......Page 313
9.5 Exercises ......Page 317
9.6 Derivatives of functions defined implicitly ......Page 319
9.7 Worked examples ......Page 323
9.8 Exercises ......Page 327
9.9 Maxima, minima, and saddle points ......Page 328
9.10 Second-order Taylor formula for scalar fields ......Page 333
9.11 The nature of a stationary point determined by the eigenvalues of the Hessian matrix ......Page 335
9.12 Second-derivative test for extrema of functions of two variables ......Page 337
9.13 Exercises ......Page 338
9.14 Extrema with constraints. Lagrange's multipliers ......Page 339
9.15 Exercises ......Page 343
9.16 The extreme-value theorem for continuous scalar fields ......Page 344
9.17 The small-span theorem for continuous scalar fields (uniform continuity) ......Page 346
10.2 Paths and line integrals ......Page 348
10.3 Other notations for line integrals ......Page 349
10.4 Basic properties of line integrals ......Page 351
10.6 The concept of work as a line integral ......Page 353
10.7 Line integrals with respect to arc length ......Page 354
10.8 Further applications of line integrals ......Page 355
10.9 Exercises ......Page 356
10.10 Open connected sets. Independence of the path ......Page 357
10.11 The second fundamental theorem of calculus for line integrals ......Page 358
10.12 Applications to mechanics ......Page 360
10.13 Exercises ......Page 361
10.14 The first fundamental theorem of calculus for line integrals ......Page 362
10.15 Necessary and sufficient conditions for a vector field to be a gradient ......Page 364
10.16 Necessary conditions for a vector field to be a gradient ......Page 365
10.17 Special methods for constructing potential functions ......Page 367
10.18 Exercises ......Page 370
10.19 Applications to exact differential equations of first order ......Page 371
10.20 Exercises ......Page 374
10.21 Potential functions on convex sets ......Page 375
11.2 Partitions of rectangles. Step functions ......Page 378
11.3 The double integral of a step function ......Page 380
11.5 Upper and lower double integrals ......Page 382
11.6 Evaluation of a double integral by repeated one-dimensional integration ......Page 383
11.7 Geometric interpretation of the double integral as a volume ......Page 384
11.8 Worked examples ......Page 385
11.9 Exercises ......Page 387
11.10 Integrability of continuous functions ......Page 388
11.11 Integrability of bounded functions with discontinuities ......Page 389
11.12 Double integrals extended over more general regions ......Page 390
11.13 Applications to area and volume ......Page 393
11.14 Worked examples 3 ......Page 94
11.15 Exercises ......Page 396
11.16 Further applications of double integrals ......Page 398
11.17 Two theorems of Pappus ......Page 401
11.18 Exercises ......Page 402
11.19 Green's theorem in the plane ......Page 403
11.20 Some applications of Green's theorem ......Page 407
11.21 A necessary and sufficient condition for a two-dimensional vector field to be a gradient ......Page 408
11.22 Exercises ......Page 410
* 11.23 Green's theorem for multiply connected regions ......Page 412
* 11.24 The winding number ......Page 414
* 11.25 Exercises ......Page 416
11.26 Change of variables in a double integral ......Page 417
11.27 Special cases of the transformation formula ......Page 421
11.28 Exercises ......Page 424
11.29 Proof of the transformation formula in a special case ......Page 426
11.30 Proof of the transformation formula in the general case ......Page 428
11.31 Extensions to higher dimensions ......Page 430
11.32 Change of variables in an «-fold integral ......Page 432
11.33 Worked examples ......Page 434
11.34 Exercises ......Page 438
12.1 Parametric representation of a surface ......Page 442
12.2 The fundamental vector product ......Page 445
12.3 The fundamental vector product as a normal to the surface ......Page 448
12.5 Area of a parametric surface ......Page 449
12.6 Exercises ......Page 454
12.7 Surface integrals ......Page 455
12.8 Change of parametric representation ......Page 457
12.9 Other notations for surface integrals ......Page 459
12.10 Exercises ......Page 461
12.11 The theorem of Stokes ......Page 463
12.12 The curl and divergence of a vector field ......Page 465
12.13 Exercises ......Page 467
12.14 Further properties of the curl and divergence ......Page 468
12.15 Exercises ......Page 472
* 12.16 Reconstruction of a vector field from its curl ......Page 473
* 12.17 Exercises ......Page 477
12.18 Extensions of Stokes'theorem ......Page 478
12.19 The divergence theorem (Gauss' theorem) ......Page 482
12.20 Applications of the divergence theorem ......Page 485
12.21 Exercises ......Page 487
Part 3. Special Topics ......Page 492
13.1 Historical introduction ......Page 494
13.2 Finitely additive set functions ......Page 495
13.3 Finitely additive measures ......Page 496
13.4 Exercises ......Page 497
13.5 The definition of probability for finite sample spaces ......Page 498
13.6 Special terminology peculiar to probability theory ......Page 500
13.8 Worked examples ......Page 502
13.9 Exercises ......Page 504
13.10 Some basic principles of combinatorial analysis ......Page 506
13.11 Exercises ......Page 510
13.12 Conditional probability ......Page 511
13.13 Independence ......Page 513
13.14 Exercises ......Page 515
13.15 Compound experiments ......Page 517
13.16 Bernoulli trials ......Page 520
13.17 The most probable number of successes in n Bernoulli trials ......Page 522
13.18 Exercises ......Page 524
13.19 Countable and uncountable sets ......Page 526
13.20 Exercises ......Page 529
13.21 The definition of probability for countably infinite sample spaces ......Page 531
13.23 Miscellaneous exercises on probability ......Page 532
14.1 The definition of probability for uncountable sample spaces ......Page 535
14.2 Countability of the set of points with positive probability ......Page 536
14.3 Random variables ......Page 537
14.4 Exercises ......Page 538
14.5 Distribution functions ......Page 539
14.6 Discontinuities of distribution functions ......Page 542
14.7 Discrete distributions. Probability mass functions ......Page 545
14.8 Exercises ......Page 548
14.9 Continuous distributions. Density functions ......Page 550
14.10 Uniform distribution over an interval ......Page 551
14.11 Cauchy's distribution ......Page 555
14.12 Exercises ......Page 557
14.13 Exponential distributions ......Page 558
14.14 Normal distributions ......Page 560
14.15 Remarks on more general distributions ......Page 564
14.16 Exercises ......Page 565
14.17 Distributions of functions of random variables ......Page 566
14.18 Exercises ......Page 567
14.19 Distributions of two-dimensional random variables ......Page 568
14.20 Two-dimensional discrete distributions ......Page 570
14.21 Two-dimensional continuous distributions. Density functions ......Page 571
14.22 Exercises ......Page 573
14.23 Distributions of functions of two random variables ......Page 575
14.24 Exercises ......Page 578
14.25 Expectation and variance ......Page 581
14.26 Expectation of a function of a random variable ......Page 584
14.27 Exercises ......Page 585
14.28 Chebyshev's inequality ......Page 587
14.29 Laws of large numbers ......Page 589
14.30 The central limit theorem of the calculus of probabilities ......Page 591
14.31 Exercises ......Page 593
Suggested References ......Page 594
15.1 Historical introduction ......Page 596
15.2 Approximations by polynomials ......Page 597
15.3 Polynomial approximation and normed linear spaces ......Page 599
15.4 Fundamental problems in polynomial approximation ......Page 600
15.5 Exercises ......Page 602
15.6 Interpolating polynomials ......Page 604
15.7 Equally spaced interpolation points ......Page 607
15.8 Error analysis in polynomial interpolation ......Page 608
15.9 Exercises ......Page 610
15.10 Newton's interpolation formula ......Page 613
15.11 Equally spaced interpolation points. The forward difference operator ......Page 615
15.12 Factorial polynomials ......Page 617
15.13 Exercises ......Page 618
15.14 A minimum problem relative to the max norm ......Page 620
15.15 Chebyshev polynomials ......Page 621
15.16 A minimal property of Chebyshev polynomials ......Page 623
15.17 Application to the error formula for interpolation ......Page 624
15.18 Exercises ......Page 625
15.19 Approximate integration. The trapezoidal rule ......Page 627
15.20 Simpson's rule ......Page 630
15.21 Exercises ......Page 635
15.22 The Euler summation formula ......Page 638
15.23 Exercises ......Page 643
Suggested References ......Page 646
Answers to exercises ......Page 647
Index ......Page 690