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Advanced Calculus: A Differential Forms Approach

Harold M. Edwards
Publication Date: 
Number of Pages: 
Modern Birkhäuser Classics
[Reviewed by
Michael Berg
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As I suppose is the case for many of us, I first encountered H. M. Edwards as the author of the wonderful book Riemann’s Zeta Function, a marvelous work of historical scholarship and an exposition of the inner life of the zeta function, with the hypothesis on center stage. Edwards wrote the book in 1974, pretty early in his career as a prolific author of books with a heavy historical dimension to them. It turns out, however, that Riemann’s Zeta Function is not his first book. The first edition of Advanced Calculus: A Differential Forms Approach, the book under review, appeared in 1969 as a somewhat unorthodox approach to undergraduate analysis: do it with an eye toward differential geometry, and, what is more, do it with differential geometry.

Well, this requires some explanation. Edwards’ approach is to take calculus proper and present in in a truly advanced way, with the emphasis placed not so much on continuity and differentiability, as is the case with most undergraduate real analysis courses, but to build things up from the foundation of the integral as a linear functional acting on differential forms. This perspective and the attendant pedagogical strategy is illustrated by the sequence in which Edwards presents his topics: the first four chapters take the reader — or the student: this is a class-room text — from the basics of the theory of integration (in the indicated extended sense) to such differential geometric mainstays as Jacobians and the implicit function theorem. Then the fifth chapter talks about differentiation in a now natural but objectively surprising way: we encounter the implicit function theorem and Lagrange multipliers, for example — not what you would generally find in an undergraduate analysis course. (On the other hand I do remember the late Robert Steinberg doing this sort of thing in the second semester of my undergraduate real analysis course, and my being very surprised: where were all the epsilons and deltas from the first semester?)

After Edwards’ fifth chapter, however, it’s back to integrals: Stokes’ theorem (done in the style of Weil, so to speak: \( \int_{\partial S}\omega = \int_S d\omega\) ) and ultimately, in “Integrals as Functions of S,” this being a discussion of the “meaning of \(\int_S \omega\) by considering it as a function of \(S\) for fixed \(\omega\),” obviously an invitation to some pretty deep mathematical thinking for a garden-variety undergraduate.

This brings us to a risk the present book poses in the ordinary scheme of things: in his “Preface to the 1994 Edition” (the book under review being a reprint thereof) Edwards states that “colleagues have sometimes expressed the opinion that the book is too difficult for the average student of advanced calculus, and is suited only to honors students.” This is a loaded caveat in today’s academic climate, of course, where we find abundant numbers of misguided and underprepared undergraduates who in point of fact require remediation even in upper division courses. Assuming a strong class, however, Edwards, nonetheless puts the burden back on the shoulders of the instructor: “I believe these colleagues think the book is difficult because it requires that they, as teachers, rethink the material and accustom themselves to a new point of view.” And this is indeed the case, of course, and does pose a challenge to all involved, the man with the chalk at the blackboard as well as the kid in the desk with his pencil and paper. But the kid had better be both well-prepared and highly motivated. Ditto for the professor.

With all this having been said this is truly an irresistible book. In 1980 Creighton Buck wrote an “Introduction” to the book, included in the 1994 (and present) edition, that includes the following: “… one becomes envious of the physicist who frosts his elementary course by references to quarks, gluons, and black holes, thereby giving his students the illusion of contact with the frontiers of research in physics … In mathematics it is much harder to bring recent research into an introductory course. This is the achievement of the author of the text before you. He has taken one of the jewels of modern mathematics — the theory of differential forms — and made this far reaching generalization of the fundamental theorem of calculus the basis of a second course in calculus … [The book] starts from the calculus of Leibniz and the Bernoullis, and moves smoothly to that of Cartan.”

The book closes with material (in the last three chapters) on Newton’s method, ODEs, harmonic functions, a little homology, flows [!], and, yes indeed, a “Further Study of Limits” which includes, e.g., discussions of uniform continuity and differentiability, compactness, Lebesgue integration (which Edwards characterizes as “a limiting case of Riemann integration”) and Banach spaces.

I think this is a wonderful book indeed. It’s not for the average course and requires an able and zealous instructor, but it’s all well worth doing, and doing well. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Constant Forms


Integration and Differentiation

Linear Algebra

Differential Calculus

Integral Calculus

Practical Methods of Solution


Further Study of Limits


Answers to Exercises