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### Summary

### Summary

This text covers most of the standard topics in multivariate calculus and part of a standard first course in linear algebra. It focuses on underlying ideas, integrates theory and applications, offers a host of pedagogical aids, and features coverage of differential forms and an emphasis on numerical methods to prepare students for modern applications of mathematics. *Covers important material that is usually omitted. *Presents more difficult and longer proofs (e.g. Proofs of the Kantorovitch theorem, the implicit function theorem) in an appendix. *Makes a careful distinction between vectors and points. *Features an innovative approach to the implicit function theorem and inverse function theorem using Newton's method. *Always emphasizes the underlying meaning - what is really going on (generally, with a geometric interpretation) - eg. The chain rule is a composition of linear transformations; the point of the implicit function theorem is to guarantee that under certain circumstances, non-linear equations have solutions. *Integrates theory and applications. *Begins most chapters with a treatment of a linear problem and then shows how the 7 methods apply to corresponding non-linear p

### Reviews 1

### Choice Review

The decade of the '50s saw the emergence of a stunningly lucid series of Princeton University seminar notes, most notably by John Milnor, recording the remarkable advances in differential geometry then being achieved by such as Atiyah, Bott, Chern, Hirzebruch, Thom, and of course Milnor himself. Not long after there appeared a draft of "Advanced Calculus" by Nickerson, Spencer, and Steenrod (NSS), though it seems never to have been truly published; this brought down to the sophomore level the understanding of those marvelous Princeton seminar notes. The appearance of NSS marked the beginning of a small but important countercurrent to the implacable flow of overweight, traditional, rote, mechanical, mainstream multivariable calculus books--one thinks fondly of Michael Spivak's Calculus on Manifolds (1965); Casper Goffman's Calculus of Several Variables (1965); Wendell H. Fleming's Functions of Several Variables (1965); John W. Woll Jr.'s Functions of Several Variables (1966); and Lynn H. Loomis and Shlomo Sternberg's Advanced Calculus (1968). The book under review is the latest stellar manifestation of this welcome trend. Like its predecessors, it deftly weaves into its exposition of multivariable calculus linear algebra, differential forms, and smooth manifolds. Unlike them, it also exploits today's calculator- and computer-aided symbolic computation and graphical display capabilities. Superb on all counts! Undergraduates through professionals. F. E. J. Linton Wesleyan University

### Table of Contents

Preface | p. xi |

Chapter 0 Preliminaries | |

0.0 Introduction | p. 1 |

0.1 Reading Mathematics | p. 1 |

0.2 Quantifiers and Negation | p. 4 |

0.3 Set Theory | p. 6 |

0.4 Functions | p. 9 |

0.5 Real Numbers | p. 17 |

0.6 Infinite Sets | p. 22 |

0.7 Complex Numbers | p. 26 |

Chapter 1 Vectors, Matrices, and Derivatives | |

1.0 Introduction | p. 33 |

1.1 Introducing the Actors: Points and Vectors | p. 34 |

1.2 Introducing the Actors: Matrices | p. 43 |

1.3 A Matrix as a Transformation | p. 59 |

1.4 The Geometry of R[superscript n] | p. 71 |

1.5 Limits and Continuity | p. 89 |

1.6 Four Big Theorems | p. 110 |

1.7 Differential Calculus | p. 125 |

1.8 Rules for Computing Derivatives | p. 146 |

1.9 Mean Value Theorem and Criteria for Differentiability | p. 154 |

1.10 Review Exercises for Chapter 1 | p. 162 |

Chapter 2 Solving Equations | |

2.0 Introduction | p. 169 |

2.1 The Main Algorithm: Row Reduction | p. 170 |

2.2 Solving Equations Using Row Reduction | p. 178 |

2.3 Matrix Inverses and Elementary Matrices | p. 186 |

2.4 Linear Combinations, Span, and Linear Independence | p. 192 |

2.5 Kernels, Images, and the Dimension Formula | p. 206 |

2.6 An Introduction to Abstract Vector Spaces | p. 224 |

2.7 Newton's Method | p. 237 |

2.8 Superconvergence | p. 257 |

2.9 The Inverse and Implicit Function Theorems | p. 264 |

2.10 Review Exercises for Chapter 2 | p. 285 |

Chapter 3 Higher Partial Derivatives, Quadratic Forms, and Manifolds | |

3.0 Introduction | p. 291 |

3.1 Manifolds | p. 292 |

3.2 Tangent Spaces | p. 316 |

3.3 Taylor Polynomials in Several Variables | p. 323 |

3.4 Rules for Computing Taylor Polynomials | p. 335 |

3.5 Quadratic Forms | p. 343 |

3.6 Classifying Critical Points of Functions | p. 353 |

3.7 Constrained Critical Points and Lagrange Multipliers | p. 360 |

3.8 Geometry of Curves and Surfaces | p. 377 |

3.9 Review Exercises for Chapter 3 | p. 394 |

Chapter 4 Integration | |

4.0 Introduction | p. 399 |

4.1 Defining the Integral | p. 400 |

4.2 Probability and Centers of Gravity | p. 415 |

4.3 What Functions Can Be Integrated? | p. 428 |

4.4 Integration and Measure Zero (Optional) | p. 435 |

4.5 Fubini's Theorem and Iterated Integrals | p. 443 |

4.6 Numerical Methods of Integration | p. 455 |

4.7 Other Pavings | p. 467 |

4.8 Determinants | p. 469 |

4.9 Volumes and Determinants | p. 485 |

4.10 The Change of Variables Formula | p. 492 |

4.11 Lebesgue Integrals | p. 505 |

4.12 Review Exercises for Chapter 4 | p. 523 |

Chapter 5 Volumes of Manifolds | |

5.0 Introduction | p. 527 |

5.1 Parallelograms and their Volumes | p. 528 |

5.2 Parametrizations | p. 532 |

5.3 Computing Volumes of Manifolds | p. 540 |

5.4 Fractals and Fractional Dimension | p. 553 |

5.5 Review Exercises for Chapter 5 | p. 555 |

Chapter 6 Forms and Vector Calculus | |

6.0 Introduction | p. 557 |

6.1 Forms on R[superscript n] | p. 558 |

6.2 Integrating Form Fields over Parametrized Domains | p. 574 |

6.3 Orientation of Manifolds | p. 579 |

6.4 Integrating Forms over Oriented Manifolds | p. 590 |

6.5 Forms and Vector Calculus | p. 602 |

6.6 Boundary Orientation | p. 614 |

6.7 The Exterior Derivative | p. 627 |

6.8 The Exterior Derivative in the Language of Vector Calculus | p. 635 |

6.9 The Generalized Stokes's Theorem | p. 642 |

6.10 The Integral Theorems of Vector Calculus | p. 651 |

6.11 Potentials | p. 658 |

6.12 Review Exercises for Chapter 6 | p. 664 |

Appendix A Some Harder Proofs | |

A.0 Introduction | p. 669 |

A.1 Arithmetic of Real Numbers | p. 669 |

A.2 Cubic and Quartic Equations | p. 673 |

A.3 Two Extra Results in Topology | p. 679 |

A.4 Proof of the Chain Rule | p. 680 |

A.5 Proof of Kantorovich's theorem | p. 682 |

A.6 Proof of Lemma 2.8.5 (Superconvergence) | p. 688 |

A.7 Proof of Differentiability of the Inverse Function | p. 690 |

A.8 Proof of the Implicit Function Theorem | p. 693 |

A.9 Proof of Theorem 3.3.9: Equality of Crossed Partials | p. 696 |

A.10 Proof of Proposition 3.3.19 | p. 698 |

A.11 Proof of Rules for Taylor Polynomials | p. 701 |

A.12 Taylor's Theorem with Remainder | p. 706 |

A.13 Proof of Theorem 3.5.3 (Completing Squares) | p. 711 |

A.14 Geometry of Curves and Surfaces: Proofs | p. 712 |

A.15 Proof of the Central Limit Theorem | p. 718 |

A.16 Proof of Fubini's Theorem | p. 722 |

A.17 Justifying the Use of Other Pavings | p. 726 |

A.18 Existence and Uniqueness of the Determinant | p. 728 |

A.19 Rigorous Proof of the Change of Variables Formula | p. 732 |

A.20 Justifying Volume 0 | p. 739 |

A.21 Lebesgue Measure and Proofs for Lebesgue Integrals | p. 741 |

A.22 Justifying the Change of Parametrization | p. 759 |

A.23 Computing the Exterior Derivative | p. 762 |

A.24 The Pullback | p. 766 |

A.25 Proof of Stokes' Theorem | p. 771 |

Appendix B Programs | p. 783 |

B.1 Matlab Newton Program | p. 783 |

B.2 Monte Carlo Program | p. 784 |

B.3 Determinant Program | p. 786 |

Bibliography | p. 789 |

Index | p. 791 |