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

John H. Hubbard and Barbara Burke Hubbard
Matrix Editions
Publication Date: 
Number of Pages: 
[Reviewed by
Gizem Karaali
, on

This is a weird book review. It starts with a confession and a referral:

I do not believe that I can do any better than Warwick Tucker, who wrote a detailed review of the second edition of this book for the Monthly . In that review Tucker gives an exhaustive summary of the book in its earlier reincarnation. Since the third edition retains all the strengths of the second edition, readers will in fact find an accurate evaluation of the current book in Tucker's October 2003 review.

The good news is that if your institution subscribes to JSTOR or you have bought JSTOR access with your MAA membership, you can access Tucker's review online. Nonetheless, I will still attempt to provide a self contained (though admittedly a much shorter) review of the book here.

In many colleges and universities, students take a basic multivariable calculus course right after two semesters of single variable calculus. In such courses, students are not expected to have any prior knowledge of linear algebra. Therefore many calculus textbooks introduce (what some would claim is) just the right amount of vector analysis to make things work right. Nonetheless, without a full blown excursion into linear algebra and then a quick trip back to calculus recovering the derivative as a linear transformation, many parts of multivariable calculus (eg. the chain rule, the Jacobian) have to remain quite incomprehensible or at least somewhat ad hoc.

The book under review is one solution to this problem. It is clear that John H. Hubbard and Barbara Burke Hubbard have written a text for a very particular type of student, one who accepts that reading and learning mathematics will no longer be a plug-and-chug activity, one who is intrigued by connections of mathematical ideas with one another, and more specifically, one who is looking forward to learning a vast amount of mathematics. In short, the book is a guide for a pretty tough course. It can be used for a two semester sequence which integrates multivariable calculus and linear algebra quite seamlessly, and which along the way introduces mathematical proof, the all-powerful tool of mathematical thinking. However it also includes so many details (and proofs) in basic analysis, single and multivariable, that it can even be used for more advanced students in a one semester analysis course.

When reading this book, I constantly was aware of the fact that I would have benefited immensely if I gotten my hands on it when I was an undergraduate. It is clear to me that the authors have put their hearts and souls into this project. The book has many details sprinkled in, many anecdotes, many personal opinions about how one does mathematics; any student interested in mathematics would find it a valuable experience to even flip through its pages randomly. I was especially excited about the last chapter where the natural framework of differential forms is developed and applied to the theory of electromagnetism.

The second edition was definitely more than a good enough book; one may ask why there is a third edition at all or why a reader should consider the third edition instead of the second. There are two answers. The first is of course that the authors have made many improvements. In fact several earlier typos are corrected and, more substantively, three new sections are added to the text. One is on eigenvectors and eigenvalues, which are now introduced in a way independent of the determinant and thus made more amenable to computation. This agrees with the book's general emphasis on computationally effective algorithms. The second new section includes rules for computing Taylor polynomials, and the third, mentioned above, is on electromagnetism.

The second reason to consider this third edition may be quite surprising to those who are used to new editions of famous calculus texts. The third edition of Vector Calculus, Linear Algebra and Differential Forms, a Unified Approach is published by a small publishing company, Matrix Editions, which specializes in books of "serious mathematics, written with the reader in mind." This book is certainly basic mathematics written for a serious reader. Most significantly, the reader will be happy with the price tag: an eight hundred page textbook for less than $80 is a bargain these days. This edition is, in fact, significantly cheaper than the second edition.

I was very impressed with the depth, clarity and ambition of this book. It respects its readers, it assumes that they are intelligent and naturally curious about beautiful mathematics. Then it provides them with all the tools necessary to learn multivariable calculus, linear algebra and basic analysis. I definitely recommend the book to anyone who is planning to teach or learn multivariable calculus.

Gizem Karaali is assistant professor of Mathematics at Pomona College.

Preface           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   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     
          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  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   
          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    

           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

           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    Electromagnetism  633
           6.10  The generalized Stokes's theorem  646
           6.12  The integral theorems of vector calculus   655
           6.13  Potentials    663
           6.13  Review exercises for chapter 6   668

 Appendix:  Analysis         
           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