CHAPTER 0 PRELIMINARIES 1
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 32
1.0 Introduction 32
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
1.4 The geometry of Rn 67
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 161
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 283
3.0 Introduction 283
3.1 Manifolds 284
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 343
3.7 Constrained critical points and Lagrange multipliers 350
3.8 Geometry of curves and surfaces 368
3.9 Review exercises for chapter 3 386
CHAPTER 4 INTEGRATION 391
4.0 Introduction 391
4.1 Defining the integral 392
4.2 Probability and centers of gravity 407
4.3 What functions can be integrated? 421
4.4 Measure zero 428
4.5 Fubini's theorem and iterated integrals 436
4.6 Numerical methods of integration 448
4.7 Other pavings 459
4.8 Determinants 461
4.9 Volumes and determinants 476
4.10 The change of variables formula 483
4.11 Lebesgue integrals 495
4.12 Review exercises for chapter 4 514
CHAPTER 5 VOLUMES OF MANIFOLDS 518
5.0 Introduction 518
5.1 Parallelograms and their volumes 519
5.2 Parametrizations 523
5.3 Computing volumes of manifolds 530
5.4 Integration and curvature 543
5.5 Fractals and fractional dimension 545
5.6 Review exercises for chapter 5 547
CHAPTER 6 FORMS AND VECTOR CALCULUS 549
6.0 Introduction 549
6.1 Forms on Rn 550
6.2 Integrating form fields over parametrized domains 565
6.3 Orientation of manifolds 570
6.4 Integrating forms over oriented manifolds 581
6.5 Forms in the language of vector calculus 592
6.6 Boundary orientation 604
6.7 The exterior derivative 617
6.8 Grad,curl,div,and all that 624
6.9 Elctromagnetism 633
6.10 The generalized Stokes's theorem 646
6.11 The integral theorems of vector calculus 655
6.12 Potentials 663
6.13 Review exercises for chapter 6 668
APPENDIX：ANALYSIS 673
A.0 Introduction 673
A.1 Arithmetic of real numbers 673
A.2 Cubic and quartic equations 677
A.3 Two results in topology：nested compact sets and Heine-Borel 682
A.4 Proof of the chain rule 683
A.5 Proof of Kantorovich's theorem 686
A.6 Proof of lemma 2.9.5(superconvergence) 692
A.7 Proof of differentiability of the inverse function 694
A.8 Proof of the implicit function theorem 696
A.9 Proving equality of crossed partials 700
A.10 Functions with many vanishing partial derivatives 701
A.11 Proving rules for Taylor polynomials；big O and little o 704
A.12 Taylor's theorem with remainder 709
A.13 Proving theorem 3.5.3(completing squares) 713
A.14 Geometry of curves and surfaces：proofs 714
A.15 Stirling's formula and proof of the central limit theorem 720
A.16 Proving Fubini's theorem 724
A.17 Justifying the use of other pavings 727
A.18 Results concerning the determinant 729
A.19 Change of variables formula：a rigorous proof 734
A.20 Justifying volume 0 740
A.21 Lebesgue measure and proofs for Lebesgue integrals 742
A.22 Justifying the change of parametrization 760
A.23 Computing the exterior derivative 765
A.24 The pullback 769
A.25 Proving Stokes's theorem 774
BIBLIOGRAPHY 788
PHOTO CREDITS 790
INDEX 792