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

John H. Hubbard , Barbara Burke Hubbard
4.6 out of 5 stars  See all reviews (5 customer reviews)

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Book Description

Sept. 7 2001 0130414085 978-0130414083 2
For an undergraduate course in Vector Calculus.
Using a dual presentation that is rigorous and comprehensive—yet exceptionaly student-friendly in approach—this text covers most of the standard topics in multivariate calculus and part of a standard first course in linear algebra. It focuses in 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.

Features and Benefits

NEW - Revised and expanded content - Including new discussions of functions; complex numbers; closure, interior, and boundary; orientation; forms restricted to vector spaces; and expanded discussions of subsets and subspaces of R^n; probability; change of basis matrix, and more, makes text now a little easier and clearer.
NEW - Approximately 875 exercises - 270 more exercises that the previous edition, reinforces students understanding of the material.
NEW - Approximately 294 examples - 50 more examples that the previous edition, deepens students understanding of concepts and theorems by allowing them to see each step of the process.
NEW - 50 more figures and tables than the previous edition, makes text a bit easier.
NEW - Student Solutions Manual, gives students detailed solutions to several hundred exercises and provides a source of additional examples.
Unified approach to vector calculus, linear algebra, and differential forms, gives students a better understanding of all three areas and shows how they all relate to each other.
Unique treatment of differential forms, shows students how differential forms in three dimensions translate into the language of vector calculus.
Stresses computationally effective algorithms—Both for computations and for underlying theory, illustrates for students how mathematics is done today.
More difficult and longer proofs in the appendix, allows more advanced students to use the book at a higher level and beginning students to focus on statements and becoming at ease with techniques rather than being intimidated by the technical details of proofs.
Emphasis on the correspondence between different mathematical languages, including algebra and geometry.
Presents material as more intuitive and conceptual.
Integrates theory and application.
Emphasizes computationally effective algorithms and proves theorems by showing that those algorithms really work.
Begins most chapters with a treatment of a linear problem, shows students how the methods apply to “corresponding” non-linear problems.
Instant exercises with solutions in footnotes, encourages students to read the text actively.

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From the Author

Several readers have complained about the lack of a student solution manual. One now exists, published by Matrix Editions. Errata for the book are posted on the book web site (URL given in the book). The most recent posting was Feb. 29, 2002. Readers who wish to be notified by e-mail when new errata are posted can sign up via the web site or e-mail the authors (address given in the book).

What's new in the second edition (the one with the pale yellow cover now being sold):

The main change is that we introduce a new approach to Lebesgue integration. In addition, the second edition has approximately 270 additional exercises and 50 additional examples. We have added pictures of mathematicians and more historical notes. There are now end-of-section exercises, as well as review exercises for Chapters 1--6. Some useful formulas are listed on the back cover.

More difficult material from Chapter 0 was moved to the Appendix. The inverse and implicit function theorems have been rewritten. In Chapter 3 we simplified the definition of a manifold, and we now begin with the general case and discuss curves and surfaces as examples. Similarly, in Chapter 5, we eliminated the separate sections on arc length and surface area; we now have one section on volume of manifolds.

In Chapter 6, we rewrote the discussion of orientation and changed the definition of a piece-with-boundary of a manifold, to make it both simpler and more inclusive.

From the Inside Flap

... The numerical interpretation ... is however necessary . ... So long
as it is not obtained, the solutions may be said to remain incomplete and
useless, and the truth which it is proposed to discover is no less hidden
in the formulae of analysis than it was in the physical problem itself.

– Joseph Fourier, The Analytic Theory of Heat

Chapters 1 through 6 of this book cover most of the standard topics in multivariate calculus and a substantial part of a standard first course in linear algebra. If, in addition, one teaches the proofs in Appendix A, the book can be used as a textbook for a course in analysis.

The organization and selection of material differs from the standard approach in three ways.

First, we believe that at this level linear algebra should be more a convenient setting and language for multivariate calculus than a subject in its own right. The guiding principle of this unified approach is that locally, a nonlinear function behaves like its derivative. Thus whenever we have a question about a nonlinear function we will answer it by looking carefully at a linear transformation: its derivative. In this approach, everything learned about linear algebra pays off twice: first for understanding linear equations, then as a tool for understanding nonlinear equations.

We discuss abstract vector spaces in Section 2.6, but the emphasis is on R', as we believe that most students find it easiest to move from the concrete to the abstract.

Second, we emphasize computationally effective algorithms, and prove theorems by showing that those algorithms really work. We feel this better reflects the way this mathematics is used today, in both applied and in pure mathematics. Moreover, it can be done with no loss of rigor.

For linear equations, row reduction (the practical algorithm) is the central tool from which everything else follows, and we use row reduction to prove all the standard results about dimension and rank. For nonlinear equations, the cornerstone is Newton's method, the best and most widely used method for solving nonlinear equations; we use it both as a computational tool and in proving the inverse and implicit function theorems. We include a section on numerical methods of integration, and we encourage the use of computers both to reduce tedious calculations and as an aid in visualizing curves and surfaces.

Third, we use differential forms to generalize the fundamental theorem of calculus to higher dimensions. The great conceptual simplifications gained by doing electromagnetism in the language of forms is a central motivation for using forms, and we will apply the language of forms to electromagnetism in a subsequent volume.

In our experience, differential forms can be taught to freshmen and sophomores, if forms are presented geometrically, as integrands that take an oriented piece of a curve, surface, or manifold, and return a number. We are aware that students taking courses in other fields need to master the language of vector calculus, and we devote three sections of Chapter 6 to integrating the standard vector calculus into the language of forms.

Chapter 0 is intended as a resource. Students should not feel that they need to read it before beginning Chapter 1. Another resource is the inside back cover, which lists some useful formulas.

Numbering of theorems, examples, and equations

Theorems, lemmas, propositions, corollaries, and examples share the same numbering system: Proposition 2.3.8 is not the eighth proposition of Section 2.3; it is the eighth numbered item of that section. We often refer back to theorems, examples, and so on, and hope this numbering will make them easier to find.

Figures and tables share their own numbering system; Figure 4.5.2 is the second figure or table of Section 4.5. Virtually all displayed equations are numbered, with the numbers given at right; Equation 4.2.3 is the third equation of Section 4.2. When an equation is displayed a second time, it keeps its original number, but the number is in parentheses.

We use a symbol to mark the end of an example or remark, and to mark the end of a proof.


Mathematical notation is not always uniform. For example, A can mean the length of a matrix A or the determinant of A. Different notations for partial derivatives also exist. This should not pose a problem for readers who begin at the beginning and end at the end, but for those who are using only selected chapters, it could be confusing. Notations used in the book are listed on the front inside cover, along with an indication of where they are first introduced.


Exercises are given at the end of each section; chapter review exercises are given at the end of each chapter, except Chapter 0. Exercises range from very easy exercises intended to make the student familiar with vocabulary, to quite difficult exercises. The hardest exercises are marked with an asterisk (in rare cases, two asterisks). A student solution manual is planned, with complete solutions to odd-numbered exercises.

What's new in the second edition

While anyone familiar with the first edition will recognize the second, there have been many changes, large and small. Perhaps the biggest change is that we now treat Lebesgue integration (Section 4.11). Other major ones include:

  1. More exercises, more examples, and more figures, including approximately 270 additional exercises and 50 additional examples. We have also added pictures of mathematicians.
  2. There are now end-of-section exercises, as well as review exercises for Chapters 1-6.
  3. More difficult material from Chapter 0 was moved to the Appendix.
  4. Chapter 0 now includes a new section on functions; some material from Section 1.3 was moved to this section. The section on complex numbers was expanded.
  5. The inverse and implicit function theorems have been rewritten.
  6. In Chapter 3 we simplified the definition of a manifold, and we now begin with the general case and discuss curves and surfaces as examples, Similarly, in Chapter 5, we eliminated the separate sections on arc length and surface area; we now have one section on volume of manifolds.
  7. In Chapter 6, we rewrote the discussion of orientation and changed the definition of a piece-with-boundary of a manifold, to make it both simpler and more inclusive.
  8. Some useful formulas are listed on the back cover.
Different ways to use this book

This book can be used at different levels of rigor. Chapters 1 through 6 contain material appropriate for a course in linear algebra and multivariate calculus. Appendix A contains the technical, rigorous underpinnings appropriate for a course in analysis. It includes proofs of those statements not proved in the main text, and a painstaking justification of arithmetic.

Most of the proofs included in this appendix are more difficult than the proofs contained in the main text, but difficulty was not the only criterion; many students find the proof of the fundamental theorem of algebra (Section 1.6) quite difficult. But we find this proof qualitatively different from the proof of the Kantorovich theorem, for example. A professional mathematician who has understood the proof of the fundamental theorem of algebra should be able to reproduce it. A professional mathematician who has read through the proof of the Kantorovich theorem, and who agrees that each step is justified, might well want to refer to printed notes in order to reproduce it. In this sense, the first proof is more conceptual, the second more technical.

Following is a brief description of different courses that could be taught using this book.

One-year courses

At Cornell University this book is used for the honors courses Math 223 (fall semester) and 224 (spring semester), for students who have studied one-variable calculus. Students are expected to have a 5 on the Advanced Placement BC Calculus exam, or the equivalent. When John Hubbard teaches the course, he typically gets to the middle of Chapter 4 in the first semester, skipping Section 3.8 on the geometry of curves and surfaces and Section 4.4 on measure 0. In the second semester he gets to the end of Chapter 6 and goes on to teach some of the material that will appear in a sequel volume, in particular differential equations.

Eventually, he would like to take three semesters to cover Chapters 1-6 of the current book and the material of the second book (referred to as "Volume 2" throughout this text):

  • Chapter 7: Differential Equations
  • Chapter 8: Inner Products and Signal Processing
  • Chapter 9: Linear Differential Equations with Constant Coefficients
  • Chapter 10: Differential Forms: Advanced Topics
  • Chapter 11: Inner Products and Forms
  • Chapter 12: Electromagnetism

Another approach would be to spend a year on Chapters 1-6. This could be done at different levels of difficulty. Some students may need to review material in Chapter 0; others may be able to include some of the proofs in the appendix.

Semester courses

(1) A one-semester course for students who have studied neither linear algebra V nor multivariate calculus.

For such a course, we suggest covering only the first four chapters, omitting the parts marked "optional" (the part of Section 2.8 concerning a stronger version of the Kantorovich theorem, and Section 4.4 on measure 0). Other topics that can be omitted include

- The proof of the fundamental theorem of algebra in Section 1.6,
- The discussion of criteria for differentiability in Section 1.9,
- Section 3.8 on the geometry of curves and surfaces,
- The proof of theorem 4.9.1,
- The discussion in Section 4.11 on Fourier and Laplace transforms.

Sections 4.2 (integrals. and probability) and 4.6 (numerical methods of integration) could also be skipped, but we feel these topics are generally given too little attention. If Section 4.2 is skipped, then one should also skip the discussion of Monte Carlo methods in Section 4.6.

(2) A course for students who have had some exposure to either linear algebra or multivariate calculus, but who are not ready for a course in analysis.

We used an earlier version of this text with students who had taken a course in linear algebra, and feel they gained a great deal from seeing how linear algebra and multivariate calculus mesh. Such students could be expected to cover Chapters 1-6, possibly omitting some of material, as discussed above. For a less fast-paced course, the book could also be covered in a year, possibly including some proofs from the appendix.

(3) A one-semester analysis course.

In one semester one could hope to cover all six chapters and some or most of the proofs in Appendix A. This could be done at varying levels of difficulty; students might be expected to follow the proofs, for example, or they might be expected to understand them well enough to construct similar proofs.

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Customer Reviews

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Most helpful customer reviews
5.0 out of 5 stars A Must Text May 14 2004
I have used this book to teach gifted high school students about the following topics: the implicit function theorem, manifolds, and differential forms. With the Hubbards' approach, even students without a course in linear algebra actually get it! Not only do they understand the material, but they also become amazingly enthusiastic when they begin to see the unifying effect of understanding differential forms.
This is the only text that I have seen that really makes forms clear. It does so by taking the time to carefully, but rigorously, explain them in a "classical" setting. One of the reasons forms are so difficult to grasp is that while some things, such as the exterior derivative and the work form of a function, can be seen as natural objects (when explained well), the connection between these objects and calculating with forms using coordinates is not so easy to make clear. The Hubbards' do make these ideas clear - even when presenting topics as hard as orientation.
Unfortunately, most of us had to wait till graduate school to see forms - usually, in a more abstract setting. By then, we probably didn't have time to sit, calculate, and make clear connections. This text makes that later transition, for those in math, much easier. It also makes physics easier. The Hubbards' make that point by showing that the electric field shouldn't really be a field, but a two form. Any book that lets one explain that - and much more - to high school students, which I do, should be a part of every multivariable calculus course.
Finally, I should note that this book contains much, much more than manifolds, the implicit function theorem and differential forms. But, even if that were all it contained, it would fully be worth the price.
In summary, this book opens the door to new worlds that most students never get to see clearly. What a present to us all.
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5.0 out of 5 stars At last - A great book on elementary mathematics Feb. 14 2002
By A Customer
This book is unique in several ways: it covers an immense amount of material, much of which is never presented in books aimed at this level. The underlying idea of the authors is to present constructive proofs, which has the great benefit of providing the reader with the ability to actually compute quantities appearing in the theorems. As an example, the Inverse Function Theorem is proved using Newton's method, which relies on Kantorovich's Theorem, and thus actually gives an explicit size of the domain on which the inverse exists. The book also contains a very nice section on Lebesgue integrals, a topic which is usually reserved for graduate level courses. The construction is to my knowledge completely new, and does not rely on sigma-algebras, but utilizes only elementary mathematics. Another nice feature is that the book considers abstract spaces at an early stage. Thus the reader is presented with the idea of computing derivatives of functions acting on e.g. matrix-spaces, as opposed to the usual Euclidian spaces. The concluding treatment on differential forms brings a lot of the introduced ideas together and completes the picture by a thorough treatment on integration over manifolds.
This book can be studied at several levels. For a first year honours course, one may skip the trickiest proofs, which appear in the appendix. More advanced readers may choose to study constructions and details of selected theorems and proofs. Anyone who buys this book will have a solid companion for many years ahead.
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5.0 out of 5 stars A beautiful book for undergrads and grads alike Feb. 21 2002
By A Customer
Although I am a graduate student in Mathematics, I found
Hubbard's "undergraduate" text to be extremely helpful.
Hubbard combines an intuitive heuristic approach appropriate
for undergraduates with a thoroughly rigorous set of proofs
appropriate for graduate students. I found his discussion of
differential forms particularly helpful. He provides an
excellent intuitive motivation for the definitions, and then
he follows this with a mathematically sound treatment of the
topic. This is a much nicer approach than one will find in
texts such as Rudin's Principals of Mathematical Analysis.
I highly recommend Hubbard's book to anyone wishing to learn
differential forms.
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5.0 out of 5 stars 2nd edition much improved May 15 2002
By A Customer
I've read sections of both the first & second editions and the second has numerous minor changes that make it a much better book. The changes are not major--the content and order are almost identical. However, places where the explanations were unclear or difficult frequently have new diagrams or helpful comments in the margins. A few topics that were too difficult or digressions have been moved to appendices or omitted. It remains a challenging book, intended for honors students, but is now a reasonable alternative to Apostol or a sequel to Spivak.
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3.0 out of 5 stars notation March 7 2004
this guy tells u a story before he gives you another story in his proof of theorem, he notation is very hard to follow. His proofs are ugly. is there a better advanced calculus book?
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