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3rd edition 

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

John Hubbard and Barbara Burke Hubbard 

book cover

There is now a 4th edition of this book

Vector Calculus 4th edition

3rd edition
ISBN 9780971576636

802 pages, hardcover, smythe-sewn binding, June 2007

Course adoptions at 
Cal State East Bay, Cornell University, Harvard University, the Harvard-Westlake School,the Hotchkiss School, Indiana University, Indiana University-Purdue University Indianapolis, UC San Diego, the University of Minnesota, the University of Montana, the University of Chicago Laboratory Schools, the University of South Dakota, Washington University in St. Louis, the Wheeler School

MAA Review: "When reading this book, I constantly was aware of the fact that I would have benefited immensely if I had gotten my hands on it when I was an undergraduate... 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."  Gizem Karaali, assistant professor of Mathematics at Pomona College  Read entire review

Praise from readers:

"I've begun reading the book and have quickly come to truly love its approach.  As a high school calculus teacher, I've read many a linear algebra text and many multivariate calculus books as well.  A former student of mine, currently an undergraduate ... at Harvard, recommended yours... and it has lived up to all the praise he has heaped upon it.
I truly value the approach you take to try to foster true understanding (including the amazingly helpful margin notes that dot each chapter), the obvious deep thought you put into the order in which all the material is presented, and the clear desire it seems the authors have to truly share their expertise in the subject in a way most likely for the target audience to fully benefit from it." 
Ross Lipsky
Valley Stream South High School Mathematics Department

"Thank you for this gold mine! (and for the cheap price too, especially for European people!)"  Marc Chambon, graduate student in inorganic chemistry and chemical engineering, Université Via Domitia, Perpignan, France

"I am having a great time reading the third edition of your book (and attempting all the problems).  It's quite user friendly without being sappy... I particularly enjoyed problem A1.5 - I thought I'd never see how Peano did it.   Part c. is worth the answer book alone." Lewis Robinson, retired M. D. indulging a lifelong taste for mathematics

Preface (excerpt in html, with link to complete preface in pdf)

To order (for books shipped to the United States)

To order (for books shipped to other countries)

Student Solution Manual for 3rd edition

Math programs used in the book

Table of contents (in html)

Look inside this book (sample pages, mostly in pdf)

Errata (latest posting March 23, 2008)

Readers praise first two editions

"Superb on all counts" - review in CHOICE (review of 1st edition)

"A real gem" - review of 2nd edition, MAA Monthly

3 new sections:

  • Eigenvectors and eigenvalues Section 2.7 on eigenvectors, eigenvalues, and diagonalization  bypasses the determinant; a few pages in chapter 4 connect this discussion with determinants and the characteristic function.
  • Integration and curvature Section 5.4 discusses the Gauss map.
  • Electromagnetism and differential forms (Section 6.9)

Other changes compared to the 2nd edition:

  • New examples  Notably an extended example on checking boundary points when looking for critical points of functions
  • Uniqueness of row echelon form The first and second editions left the proof of uniqueness in theorem 2.1.8 as an exercise. Now this important result is in the text.
  • New exercises Including an exercise proving a version of Kantorovich's theorem with a weaker continuity condition on the derivatives; this has implications for the inverse and implicit function theorems.
  • New proof of the generalized Stokes's theorem
  • Programs posted on web No more laborious typing! The three programs used in the book  are available for cut-and-paste at math programs.