Advanced Calculus

A Geometric View

  • James J. Callahan

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

Introduction

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.

Keywords

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

  • James J. Callahan
    • 1
  1. 1., Mathematics and Statistics DepartmentSmith CollegeNorthamptonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-7332-0
  • Copyright Information Springer Science+Business Media, LLC 2010
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4419-7331-3
  • Online ISBN 978-1-4419-7332-0
  • Series Print ISSN 0172-6056
  • About this book