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Advanced Calculus: A Geometric View

James J. Callahan
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

The notion of a course in “advanced calculus” has changed considerably since the days of the classic textbooks of Taylor and Buck, published in the 1950s. In one direction, the course has evolved to provide a rigorous basis for calculus and thus toward introductory real analysis. In another, it has transmuted into calculus on manifolds. Yet there is still a place in many curricula for a text that concentrates on the study of calculus in two and three dimensions. The author of this book sees an opportunity to bring back a more geometric, visual and physically-motivated approach to the subject.

Although the table of contents doesn’t look so very different from comparable texts, this book has a much different feel to it. This begins in the first chapter where we see a new spin on substitution — change of variables — in ordinary single variable integration. The author describes two different kinds of substitution that naturally give rise to the notions of pullback and push-forward. The theme of change of variables formulas in integrals, seen particularly in its geometric aspects, is an important thread throughout the book. Local linearity, viewed analytically and geometrically, is a closely related motif.

The book contains much more material than could be reasonably treated in a one semester course. The author suggests one possible program includes introductory material up to a complete treatment of the derivative followed by double integrals, surface integrals and Stokes theorem. Alternatively, a course could follow the introductory material with the chapters on inverse and implicit function theorems, perhaps as prelude to the study of differentiable manifolds.

Notable features of the book include an extended treatment of Morse’s lemma (with proof), a lengthy discussion of Jordan content in the context of double integrals, and a physically-motivated (Feynman-inspired) discussion of the curl based on fluid flow.

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.

This is probably not a text for the average second year advanced calculus student. The level of sophistication is just too high, with an uncompromising, often fairly subtle exposition. Strong students, however, are likely to be attracted by the approach and the serious meaty content.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

1 Starting Points.-1.1 Substitution.- Exercises.- 1.2 Work and path integrals.- Exercises.- 1.3 Polar coordinates.- Exercises.- 2 Geometry of Linear Maps.- 2.1 Maps from R2 to R2.- Exercises.- 2.2 Maps from Rn to Rn.- Exercises.- 2.3 Maps from Rn to Rp, n 6= p.- Exercises.- 3 Approximations.- 3.1 Mean-value theorems.- Exercises.- 3.2 Taylor polynomials in one variable.- Exercises.- 3.3 Taylor polynomials in several variables.- Exercises.- 4 The Derivative.- 4.1 Differentiability.- Exercises.- 4.2 Maps of the plane.- Exercises.- 4.3 Parametrized surfaces.- Exercises.- 4.4 The chain rule.- Exercises.- 5 Inverses.- 5.1 Solving equations.- Exercises.- 5.2 Coordinate Changes.- Exercises.- 5.3 The Inverse Function Theorem.- Exercises.- 6 Implicit Functions.- 6.1 A single equation.- Exercises.- 6.2 A pair of equations.- Exercises.- 6.3 The general case.- Exercises.- 7 Critical Points.- 7.1 Functions of one variable.- Exercises.- 7.2 Functions of two variables.- Exercises.- 7.3 Morse’s lemma.- Exercises.- 8 Double Integrals.- 8.1 Example: gravitational attraction.- Exercises.- 8.2 Area and Jordan content.- Exercises.- 8.3 Riemann and Darboux integrals.- Exercises.- 9 Evaluating Double Integrals.- 9.1 Iterated integrals.- Exercises.- 9.2 Improper integrals.- Exercises.- 9.3 The change of variables formula.- 9.4 Orientation.- Exercises.- 9.5 Green’s Theorem.- Exercises.- 10 Surface Integrals.- 10.1 Measuring flux.- Exercises.- 10.2 Surface area and scalar integrals.- Exercises.- 10.3 Differential forms.- Exercises.- 11 Stokes’ Theorem.- 11.1 Divergence.- Exercises.- 11.2 Circulation and Vorticity.- Exercises.- 11.3 Stokes’ Theorem.- 11.4 Closed and Exact Forms.- Exercises