Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach

Title......Page 1
Contents......Page 2
Preface......Page 6
0.1 Reading Mathematics......Page 12
0.2 How to Negate Mathematical Statements......Page 15
0.3 Set Theory......Page 16
0.4 Real Numbers......Page 17
0.5 Infinite Sets and Russell\'s Paradox......Page 23
0.6 Complex Numbers......Page 25
0.7 Exercises for Chapter 0......Page 31
1.0 Introduction......Page 38
1.1 Introducing the Actors: Vectors......Page 39
1.2 Introducing the Actors: Matrices......Page 46
1.3 A Matrix as a Transformation......Page 57
1.4 The Geometry of Rn......Page 69
1.5 Convergence and Limits......Page 83
1.6 Four Big Theorems......Page 100
1.7 Differential Calculus......Page 111
1.8 Rules for Computing Derivatives......Page 126
1.9 Criteria for Differentiability......Page 131
1.10 Exercises for Chapter 1......Page 138
2.0 Introduction......Page 158
2.1 The Main Algorithm: Row Reduction......Page 159
2.2 Solving Equations Using Row Reduction......Page 165
2.3 Matrix Inverses and Elementary Matrices......Page 171
2.4 Linear Combinations, Span, and Linear Independence......Page 176
2.5 Kernels and Images......Page 188
2.6 Abstract Vector Spaces......Page 200
2.7 Newton\'s Method......Page 208
2.8 Superconvergence......Page 222
2.9 The Inverse and Implicit Function Theorems......Page 228
2.10 Exercises for Chapter 2......Page 242
3.0 Introduction......Page 260
3.1 Curves and Surfaces......Page 261
3.2 Manifolds......Page 277
3.3 Taylor Polynomials in Several Variables......Page 286
3.4 Rules for Computing Taylor Polynomials......Page 296
3.5 Quadratic Forms......Page 301
3.6 Classifying Critical Points of Functions......Page 309
3.7 Constrained Critical Points and Lagrange Multipliers......Page 315
3.8 Geometry of Curves and Surfaces......Page 327
3.9 Exercises for Chapter 3......Page 343
4.0 Introduction......Page 362
4.1 Defining the Integral......Page 363
4.2 Probability and Integrals......Page 373
4.3 What Functions Can Be Integrated?......Page 383
4.4 Integration and Measure Zero (Optional)......Page 391
4.5 Fubini\'s Theorem and Iterated Integrals......Page 398
4.6 Numerical Methods of Integration......Page 406
4.7 Other Pavings......Page 415
4.8 Determinants......Page 416
4.9 Volumes and Determinant......Page 431
4.10 The Change of Variables Formula......Page 437
4.11 Improper Integrals......Page 447
4.12 Exercises for Chapter 4......Page 460
5.0 Introduction......Page 480
5.1 Parallelograms and their Volumes......Page 481
5.2 Parametrizations......Page 484
5.3 Arc Length......Page 490
5.4 Surface Area......Page 493
5.5 Volume of Manifolds......Page 499
5.6 Fractals and Fractional Dimension......Page 502
5.7 Exercises for Chapter 5......Page 503
6.0 Introduction......Page 510
6.1 Forms as Integrands over Oriented Domain......Page 511
6.2 Forms on Rn......Page 512
6.3 Integrating Form Fields over Parametrized Domains......Page 523
6.4 Forms and Vector Calcnlus......Page 527
6.5 Orientation and Integration of Form Fields......Page 536
6.6 Boundary Orientation......Page 547
6.7 The Exterior Derivative......Page 555
6.8 The Exterior Derivative in the Language of Vector Calculus......Page 561
6.9 Generalized Stokes\'s Theorem......Page 567
6.10 The Integral Theorems of Vector Calculus......Page 574
6.11 Potentials......Page 579
6.12 Exercises for Chapter 6......Page 584
A.1 Proof of the Chain Rule......Page 600
A.2 Proof of Kantorovitch\'s theorem......Page 603
A.3 Proof of Lemma 2.8.4 (Superconvergence)......Page 608
A.4 Proof of Differentiability of the Inverse Function......Page 609
A.5 Proof of the Implicit Function Theorem......Page 613
A.6 Proof of Theorem 3.3.9: Equality of Crossed Partials......Page 616
A.7 Proof of Proposition 3.3.19......Page 617
A.8 Proof of Rules for Taylor Polynomials......Page 620
A.9 Taylor\'s Theorem with Remainder......Page 624
A.10 Proof of Theorem 3.5.3 (Completing Squares)......Page 628
A.11 Proof of Propositions 3.8.12 and 3.8.13 (Frenet Formulas)......Page 630
A.12 Proof of the Central Limit Theorem......Page 633
A.13 Proof of Fubini\'s Theorem......Page 637
A.14 Justifying the Use of Other Pavings......Page 640
A.15 Existence and Uniqueness of the Determinant......Page 643
A.16 Rigorous Proof of the Change of Variables Formula......Page 646
A.17 A Few Extra Results in Topology......Page 653
A.18 Proof of the Dominated Convergence Theorem......Page 654
A.19 Justifying the Change of Parametrization......Page 659
A.20 Proof of Theorem 6.7.3 (Computing the Exterior Derivative)......Page 663
A.21 The Pullback......Page 667
A.22 Proof of Stokes\' Theorem......Page 672
A.23 Exercises......Page 676
B.1 MATLAB Newton Program......Page 680
B.2 Monte Carlo Program......Page 681
B.3 Determinant Program......Page 683
Bibliography......Page 686
Index......Page 688