Advanced Calculus of Several Variables (Dover Books on Mathematics)

This is a wonderful book. The exercises are interesting and resolvable: you do not need to be a genius. In rigorous style, it covers differential manifolds, differential forms, etc.. Buy it!

This is a clear, well motivated discussion of advanced calculus (2nd year, multivariable) using the machinery and vocabulary of the differential mapping and differential forms. This discussion takes place within Rn and all manifolds are explicitly within Rn. So this lets you get a good grounding in the ideas that are generalized in the more abstract differential geometry/Reimannian setting. There is mothing you lead to "unlearn" to adapt to the more abstract world of bundles. It gives a clear roadmap of the ideas that will need to be adapted to the new setting. This is also a great book if you just want to learn 2nd year calculus but want to see more than div,grad, curl. More suited for mathematicians and physicists than engineers.

Some people think Dover books, being cheap, ought to be bad. In fact, this Dover series specializes in "salvaging" great titles that went out of print and are of great intellectual/pedagogical value. Such is the case again for this title. Very well written. Of course, C.H. Edwards is notorious for his book on the history of calculus. Exceedingly clear. I started reading it while taking Calculus II, in search of some more elaborate perspectives. It is that clear. Chapter 1 is a brief incursion in some topological aspects. Chapter 2 directional derivatives, differentials. Ch3. Chain rule. Ch.4 Critical points. Ch. 5 MANIFOLDS (patches ?! ) and Lagrange multipliers (and this is around a bit over page 100!). Ch 6 Taylor's in one and Ch. 7 several variables. Ch 8 Classification of critical points. Part III begins with Newton's method and contraction mappings. Then goes to Multivariable mean theorem, Inverse and Implicit Mapping Theorem. Ch 4 (III) is Manifolds in Rn and finishes with higher derivatives. Part IV is Multiple Integrals, n-dimensional integrals, Riemman sums, Fubini's theorem, Change of Variables, Improper Integrals, Path Lenght and Line Integrals, Green's theorem, some applied problems, Line and Surface Integrals. Book end with Differential Forms, Stoke;s theorem, Classical Theorems of Vector Analysis, Closed and Exact Forms, Normed Vectors Spaces, Variational Calculus the Isoperimetric problem. Lots, lots of bangs for your bucks. Because of the breadth of the exposition, clarity and price, it's a must-have. You can kind of draw a parallel between this and Hubbard's Vector Calculus, Linear Algebra and Differential Forms. Both kind of span the same space. Of course, being older, it doesn't have the same computational flavor as Hubbard's (but then again, it's not really about numerical methods, is it?).

Clear, and well written. It is a great combination of linear algebra and analysis. The text is clear and the examples are illustrative.

A lot of new books have a tendency to dilute the material with really nice computer generated graphics and so called "pedagogical" methods which really don't enhance intuitive and rigorous understanding. Moreover, there are books which use physics as a way around explaining the mathematics. This book is far above them. It is a MATH book, not a science book, and has no signs of pretention. The explanations require thought but once they are understood they contribute greatly to one's appreciation. There is not doubt that a good course in algebra and calculus are required. It might even be advisable to have a some knowledge of multivariable calculus. With all these tools in hand, this volume gives much and simply asks for some patience and deligence from the student. In short, this book is about teaching mathematics in a rigorous, and comprehensive style: all proofs are given (although some are "unique") and followed by discussion. Furthermore, the exercises really are at the heart of this book. To do them is to understand.