Advanced calculus a differential forms approach djvu

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries. As I suppose is the case for many of us, I first encountered H. Edwards wrote the book in , pretty early in his career as a prolific author of books with a heavy historical dimension to them. The first edition of Advanced Calculus: A Differential Forms Approach , the book under review, appeared in 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. 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?

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. 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.

I think this is a wonderful book indeed. This affordable softcover reprint of the edition presents the diverse set of topics from which advanced calculus courses are created in beautiful unifying generalization.

The author emphasizes the use of differential forms in linear algebra, implicit differentiation in higher dimensions using the calculus of differential forms, and the method of Lagrange multipliers in a general but easy-to-use formulation.

There are copious exercises to help guide the reader in testing understanding. The chapters can be read in almost any order, including beginning with the final chapter that contains some of the more traditional topics of advanced calculus courses.

In addition, it is ideal for a course on vector analysis from the differential forms point of view. The professional mathematician will find here a delightful example of mathematical literature; the student fortunate enough to have gone through this book will have a firm grasp of the nature of modern mathematics and a solid framework to continue to more advanced studies.

The most important feature…is that it is fun—it is fun to read the exercises, it is fun to read the comments printed in the margins, it is fun simply to pick a random spot in the book and begin reading. This is the way mathematics should be presented, with an excitement and liveliness that show why we are interested in the subject. An inviting, unusual, high-level introduction to vector calculus, based solidly on differential forms.

Superb exposition: informal but sophisticated, down-to-earth but general, geometrically rigorous, entertaining but serious. Remarkable diverse applications, physical and mathematical.

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