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. James J. Callahan is currently a professor of mathematics at Smith College. Summing Up …. Upper-division undergraduate through professional collections.
Bauer, Choice, Vol. Drawings and figures are abundant and strongly support his exposition.
Catalog Record: Advanced calculus | HathiTrust Digital Library
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The edition in turn is a lightly-revised version of the Addison-Wesley edition; some clarifications were squeezed in without re-typesetting; most of the pages were unaltered, and no pages were added. So this is essentially a year-old book; is it still useful?
I say Yes. The material is still all relevant, and the subject matter has not changed a lot in the past 50 years. The present book is positioned at a comfortable level of abstraction, and no more recent books have come out to compete with it at this level. The first half of the book deals with derivatives in normed linear spaces, and the second half deals with manifolds in Banach spaces and integration in Euclidean spaces and on manifolds.
A Problems Based Course in Advanced Calculus
The pace is leisurely, with many side comments to tell us why we are doing this and where we are going. The book develops the needed background in linear spaces and topology. Integration in Euclidean spaces is done using Jordan content rather than the Lebesgue measure.
The level of abstraction is adjusted up or down throughout the book, balancing generality with ease of understanding. For example, there is a chapter on linear spaces in general, followed by a more specialized one on finite-dimensional spaces.
Manifolds are defined and their geometric properties developed in Banach spaces, but the integration work is done in finite-dimensional spaces. The book closes with two fairly detailed chapters of applications, one on potential theory and one on classical mechanics.
One minor flaw of this edition is that it is a photographic reproduction, with some of the pages being slightly faint; some of the fine lines in the type are eroded. Most of the faintness appears to have originated in the reprint, which seems to be a photographic reproduction with some modified pages. Allen Stenger is a math hobbyist and retired software developer.
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