Cover image for Vector calculus, linear algebra, and differential forms : a unified approach
Vector calculus, linear algebra, and differential forms : a unified approach
Hubbard, John H. (John Hamal), 1945 or 1946-
Publication Information:
Upper Saddle River, N.J. : Prentice Hall, [1999]

Physical Description:
xvi, 687 pages : illustrations ; 25 cm
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QA303 .H79 1999 Adult Non-Fiction Central Closed Stacks

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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

Prefacep. xi
Chapter 0 Preliminaries
0.0 Introductionp. 1
0.1 Reading Mathematicsp. 1
0.2 Quantifiers and Negationp. 4
0.3 Set Theoryp. 6
0.4 Functionsp. 9
0.5 Real Numbersp. 17
0.6 Infinite Setsp. 22
0.7 Complex Numbersp. 26
Chapter 1 Vectors, Matrices, and Derivatives
1.0 Introductionp. 33
1.1 Introducing the Actors: Points and Vectorsp. 34
1.2 Introducing the Actors: Matricesp. 43
1.3 A Matrix as a Transformationp. 59
1.4 The Geometry of R[superscript n]p. 71
1.5 Limits and Continuityp. 89
1.6 Four Big Theoremsp. 110
1.7 Differential Calculusp. 125
1.8 Rules for Computing Derivativesp. 146
1.9 Mean Value Theorem and Criteria for Differentiabilityp. 154
1.10 Review Exercises for Chapter 1p. 162
Chapter 2 Solving Equations
2.0 Introductionp. 169
2.1 The Main Algorithm: Row Reductionp. 170
2.2 Solving Equations Using Row Reductionp. 178
2.3 Matrix Inverses and Elementary Matricesp. 186
2.4 Linear Combinations, Span, and Linear Independencep. 192
2.5 Kernels, Images, and the Dimension Formulap. 206
2.6 An Introduction to Abstract Vector Spacesp. 224
2.7 Newton's Methodp. 237
2.8 Superconvergencep. 257
2.9 The Inverse and Implicit Function Theoremsp. 264
2.10 Review Exercises for Chapter 2p. 285
Chapter 3 Higher Partial Derivatives, Quadratic Forms, and Manifolds
3.0 Introductionp. 291
3.1 Manifoldsp. 292
3.2 Tangent Spacesp. 316
3.3 Taylor Polynomials in Several Variablesp. 323
3.4 Rules for Computing Taylor Polynomialsp. 335
3.5 Quadratic Formsp. 343
3.6 Classifying Critical Points of Functionsp. 353
3.7 Constrained Critical Points and Lagrange Multipliersp. 360
3.8 Geometry of Curves and Surfacesp. 377
3.9 Review Exercises for Chapter 3p. 394
Chapter 4 Integration
4.0 Introductionp. 399
4.1 Defining the Integralp. 400
4.2 Probability and Centers of Gravityp. 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 Integralsp. 443
4.6 Numerical Methods of Integrationp. 455
4.7 Other Pavingsp. 467
4.8 Determinantsp. 469
4.9 Volumes and Determinantsp. 485
4.10 The Change of Variables Formulap. 492
4.11 Lebesgue Integralsp. 505
4.12 Review Exercises for Chapter 4p. 523
Chapter 5 Volumes of Manifolds
5.0 Introductionp. 527
5.1 Parallelograms and their Volumesp. 528
5.2 Parametrizationsp. 532
5.3 Computing Volumes of Manifoldsp. 540
5.4 Fractals and Fractional Dimensionp. 553
5.5 Review Exercises for Chapter 5p. 555
Chapter 6 Forms and Vector Calculus
6.0 Introductionp. 557
6.1 Forms on R[superscript n]p. 558
6.2 Integrating Form Fields over Parametrized Domainsp. 574
6.3 Orientation of Manifoldsp. 579
6.4 Integrating Forms over Oriented Manifoldsp. 590
6.5 Forms and Vector Calculusp. 602
6.6 Boundary Orientationp. 614
6.7 The Exterior Derivativep. 627
6.8 The Exterior Derivative in the Language of Vector Calculusp. 635
6.9 The Generalized Stokes's Theoremp. 642
6.10 The Integral Theorems of Vector Calculusp. 651
6.11 Potentialsp. 658
6.12 Review Exercises for Chapter 6p. 664
Appendix A Some Harder Proofs
A.0 Introductionp. 669
A.1 Arithmetic of Real Numbersp. 669
A.2 Cubic and Quartic Equationsp. 673
A.3 Two Extra Results in Topologyp. 679
A.4 Proof of the Chain Rulep. 680
A.5 Proof of Kantorovich's theoremp. 682
A.6 Proof of Lemma 2.8.5 (Superconvergence)p. 688
A.7 Proof of Differentiability of the Inverse Functionp. 690
A.8 Proof of the Implicit Function Theoremp. 693
A.9 Proof of Theorem 3.3.9: Equality of Crossed Partialsp. 696
A.10 Proof of Proposition 3.3.19p. 698
A.11 Proof of Rules for Taylor Polynomialsp. 701
A.12 Taylor's Theorem with Remainderp. 706
A.13 Proof of Theorem 3.5.3 (Completing Squares)p. 711
A.14 Geometry of Curves and Surfaces: Proofsp. 712
A.15 Proof of the Central Limit Theoremp. 718
A.16 Proof of Fubini's Theoremp. 722
A.17 Justifying the Use of Other Pavingsp. 726
A.18 Existence and Uniqueness of the Determinantp. 728
A.19 Rigorous Proof of the Change of Variables Formulap. 732
A.20 Justifying Volume 0p. 739
A.21 Lebesgue Measure and Proofs for Lebesgue Integralsp. 741
A.22 Justifying the Change of Parametrizationp. 759
A.23 Computing the Exterior Derivativep. 762
A.24 The Pullbackp. 766
A.25 Proof of Stokes' Theoremp. 771
Appendix B Programsp. 783
B.1 Matlab Newton Programp. 783
B.2 Monte Carlo Programp. 784
B.3 Determinant Programp. 786
Bibliographyp. 789
Indexp. 791