Customer Reviews

Advanced Calculus of Several Variables (Dover Books on Mathematics)

5 star:		(6)
4 star:		(4)
3 star:		(1)
2 star:		(1)
1 star:		(0)

 

 

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?).

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.

I have not yet finished the book, but I feel I have to express my disagreement with the reviewer from Florida. The book has nice linear algebra and calculus reviews (yes, he does assume that you have a good calculus background and that reviews are just, well, reviews. This is a book about ADVANCED calculus, after all!), effective diagrams, and plenty of exercises (hundreds of them).

Overall, it is somewhat easier to read than some celebrated classics like Spivak's "Calculus on Manifolds" or Munkres's "Analysis on Manifolds."

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.

Be prepared for a lot of "connecting the dots". You should be constantly on the look-out for implied preconditions and missing proof steps. The requisite Linear Algebra can't be picked up entirely from the book. I stopped reading after section IV.3, when i realized the value i extracted from the book didn't compensate for the effort i needed to expand on gap-filling.

Definitions are, more often than not, introduced inline, rather than set apart and given a number. Some important theorems, specifically the Inverse Mapping Theorem and the Implicit Function Theorem, aren't marked distinctively. This makes for an unwieldy reference.

Some important proofs and examples are relegated to the exercise sections. Section III.5 (Higher Derivatives), for instance, is entirely exercise-based. Section II.2 cites exercise 2.5 as a counter-example to theorem 2.5 (i.e. as demonstrating how, when one of the theorem's preconditions is broken, the conclusion fails to hold), but the exercise is actually an example, not a counter-example (in other words, the exercise explores a function, which fulfills both the theorem's preconditions and conclusion).

On the up side, the proofs are concise (when not too terse) and the pace is upbeat (when not galloping). Some of the topics (specifically, Lagrange Multipliers) are dealt with in generality that i dare you find in other books.

The book could be excellent, if submitted to a thorough content-wise and graphical editorial review. As it stands, it must be supplemented with other titles, and demands more than it gives.

I don't know how good this book will be for someone who is not comfortable with proofs, because it takes a somewhat more abstract, less computational and visual approach to the subject. Nevertheless, if you already learned multivariable calculus in a computational/visual style in an early undergraduate class, and now find yourself more comfortable with the abstract approach of higher mathematics, this book will help you re-learn the concepts of multivariable calculus in the proper abstract setting. Everything clicks at reasonable reading speed so you don't need to spend a lot of time stopping to remind yourself of what he's doing. An easy, smooth read that will leave you with a good perspective on the subject. Highly recommended for the right kind of reader.

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

This book is definetely worth the price (very inexpensive). Except for the outdated notation, I would recommend this book for advanced calculus course.

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 book is a very clear exposition of the standard material covered by an advanced calculus course. I found the text neither overly abstract nor oversimplified.

The many problems helped me understand difficult points in the text that escaped me on first reading.

All in all, an excellent book.