© 2010

Advanced Calculus

A Geometric View

  • Offers important geometric approach to advanced calculus

  • Integrates text fully with 250+ illustrations

  • Treats classical advanced calculus topics Uses 2D and 3D graphics to study maps

  • Magnifies images to carry out local analysis

  • Gives visual insight into the derivative

  • Gives geometric interpretation of implicit function theorems

  • Analyzes physical meaning of divergence and curl Presents Morse's lemma and Poincaré lemma


Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. James J. Callahan
    Pages 1-28
  3. James J. Callahan
    Pages 29-70
  4. James J. Callahan
    Pages 71-104
  5. James J. Callahan
    Pages 105-150
  6. James J. Callahan
    Pages 151-184
  7. James J. Callahan
    Pages 185-218
  8. James J. Callahan
    Pages 219-268
  9. James J. Callahan
    Pages 269-316
  10. James J. Callahan
    Pages 317-386
  11. James J. Callahan
    Pages 387-448
  12. James J. Callahan
    Pages 449-514
  13. Back Matter
    Pages 151-526

About this book


With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones--the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes--the text treats other important topics in differential analysis, such as Morse's lemma and the Poincaré lemma. The ideas behind most topics can be understood with just two or three variables. This invites geometric visualization; the book incorporates modern computational tools to give visualization real power. Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps, such as the role of the derivative as the local linear approximation to a map and its role in the change of variables formula for multiple integrals. The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books. Advanced Calculus: A Geometric View is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics. Prerequisites are an introduction to linear algebra and multivariable calculus. There is enough material for a year-long course on advanced calculus and for a variety of semester courses--including topics in geometry. It avoids duplicating the material of real analysis. The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.


Algebra Derivative Green's theorem Morse's lemma Riemann and Darboux integrals Stokes' theorem calculus change of variables formula critical points derivative as linear approximation differential equation implicit functions inverse function theorem parametrized surfaces surface integrals

Authors and affiliations

  1. 1., Mathematics and Statistics DepartmentSmith CollegeNorthamptonUSA

About the authors

James J. Callahan is currently a professor of mathematics at Smith College. His previous Springer book is entitled The Geometry of Spacetime: An Introduction to Special and General Relativity. He was director of the NSF-funded Five College Calculus Project and a coauthor of Calculus in Context.

Bibliographic information


From the reviews:

“Many concepts in calculus and linear algebra have obvious geometric interpretations. … This book differs from other advanced calculus works … it can serve as a useful reference for professors. … it is the adopted course resource, its inclusion in a college library’s collection should be determined by the size and interests of the mathematics faculty. Summing Up … . Upper-division undergraduate through professional collections.” (C. Bauer, Choice, Vol. 48 (8), April, 2011)

“The author of this book sees an opportunity to bring back a more geometric, visual and physically-motivated approach to the subject. … The author makes exceptionally good use of two and three-dimensional graphics. Drawings and figures are abundant and strongly support his exposition. Exercises are plentiful and they cover a range from routine computational work to proofs and extensions of results from the text. … Strong students … are likely to be attracted by the approach and the serious meaty content.” (William J. Satzer, The Mathematical Association of America, January, 2011)

“A new geometric and visual approach to advanced calculus is presented. … The book can be useful a textbook for beginners as well as a source of supplementary material for university teachers in calculus and analysis. … the book meets a wide auditorium among undergraduate and graduate students in mathematics, physics, economics and in other fields which essentially use mathematical models. It is also very interesting for teachers and instructors in Calculus and Mathematical Analysis.” (Sergei V. Rogosin, Zentralblatt MATH, Vol. 1205, 2011)