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Matrix
Editions: Serious
mathematics,
written with the reader in mind.
Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach
ISBN 9780971576650
$84, hardcover, 818 pages, smythe-sewn binding
4th edition September 2009
In chapters 0-6 we have put in red sections where the page numbers have been changed, compared to the 3rd edition.
ContentsPreface xi
Chapter 0 Preliminaries
0.0 Introduction 1
0.1 Reading mathematics 1
0.2 Quantifiers and negation 4
0.3 Set theory 6
0.4 Functions 9
0.5 Real numbers 17
0.6 Infinite sets 22
0.7 Complex numbers 25
Chapter 1 Vectors, matrices, and derivatives
1.0 Introduction 321.4 The geometry of Rn 67
1.1 Introducing the actors: points and vectors 33
1.2 Introducing the actors: matrices 42
1.3 Matrix multiplication as a linear transformation 56
1.5 Limits and continuity 84
1.6 Four big theorems 106
1.7 Derivatives in several variables as linear transformations 120
1.8 Rules for computing derivatives 140
1.9 The mean value theorem and criteria for differentiability 148
1.10 Review exercises for chapter 1 155
Chapter 2 Solving equations
2.0 Introduction 161
2.1 The main algorithm: row reduction 162
2.2 Solving equations with row reduction 168
2.3 Matrix inverses and elementary matrices 177
2.4 Linear combinations, span, and linear independence 182
2.5 Kernels, images, and the dimension formula 195
2.6 Abstract vector spaces 211
2.7 Eigenvectors and eigenvalues 222
2.8 Newton's method 232
2.9 Superconvergence 252
2.10 The inverse and implicit function theorems 259
2.11 Review exercises for chapter 2 278
Chapter 3 Manifolds, Taylor polynomials, quadratic forms, and curvature
3.0 Introduction 284
3.1 Manifolds 285
3.2 Tangent spaces 306
3.3 Taylor polynomials in several variables 314
3.4 Rules for computing Taylor polynomials 326
3.5 Quadratic forms 334
3.6 Classifying critical points of functions 344
3.7 Constrained critical points and Lagrange multipliers 351
3.8 Geometry of curves and surfaces
3.9 Review exercises for chapter 3 389
Chapter 4 Integration
4.0 Introduction 393
4.1 Defining the integral 394
4.2 Probability and centers of gravity 409
4.3 What functions can be integrated? 423
4.4 Measure zero 430
4.5 Fubini's theorem and iterated integrals 438
4.6 Numerical methods of integration 450
4.7 Other pavings 461
4.8 Determinants 463
4.9 Volumes and determinants 478
4.10 The change of variables formula 485
4.11 Lebesgue integrals 497
4.12 Review exercises for chapter 4 516
Chapter 5 Volumes of manifolds
5.0 Introduction 520
5.1 Parallelograms and their volumes 521
5.2 Parametrizations 524
5.3 Computing volumes of manifolds 532
5.4 Integration and curvature 544
5.5 Fractals and fractional dimension 554
5.6 Review exercises for chapter 5 556
Chapter 6 Forms and vector calculus
6.0 Introduction 558
6.1 Forms on Rn 559
6.2 Integrating form fields over parametrized domains 572
6.3 Orientation of manifolds 576
6.4 Integrating forms over oriented manifolds 584
6.5 Forms in the language of vector calculus 594
6.6 Boundary orientation 606
6.7 The exterior derivative 620
6.8 Grad, curl, div, and all that 627
6.9 The generalized Stokes's theorem 635
6.10 The integral theorems of vector calculus 643
6.12 Electromagnetism 667
6.13 Potentials 667
6.13 Review exercises for chapter 6 668
Appendix: Analysis
A.0 Introduction 683
A.1 Arithmetic of real numbers 683
A.2 Cubic and quartic equations 687
A.3 Two results in topology: nested compact sets and
Heine-Borel 692
A.4 Proof of the chain rule 693
A.5 Proof of Kantorovich's theorem 696
A.6 Proof of lemma 2.9.5 (superconvergence) 702
A.7 Proof of differentiability of the inverse function 704
A.8 Proof of the implicit function theorem 706
A. 9 Proving equality of crossed partials 711
A.10 Functions with many vanishing partial derivatives 712
A.11 Proving rules for Taylor polynomials; big O and
little o 715
A.12 Taylor's theorem with remainder 720
A.13 Proving theorem 3.5.3 (completing squares) 725
A.14 Classifying constrained critical points 726 (new section)
A.15 Geometry of curves and surfaces: proofs 730
A.16 Stirling's formula and proof of the central limit theorem 735
A.17 Proving Fubini's theorem 740
A.18 Justifying the use of other pavings 742
A.19 Results concerning the determinant 745
A.20 Change of variables formula: a rigorous proof 750
A.21 Justifying volume 0 756
A.22 Lebesgue measure and proofs for Lebesgue integrals 758
A.23 Justifying the change of parametrization 776
A.24 Computing the exterior derivative 781
A.25 The pullback 785
A.26 Proving Stokes's theorem 790
Bibliography 804
Photo credits 805
Index 607
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