Functions of Several Variables (Undergraduate Texts in Mathematics) 2nd Edition by Wendell Fleming (PDF)

Description

This new edition, like the first, presents a thorough introduction to differential and integral calculus, including the integration of differential forms on manifolds. However, an additional chapter on elementary topology makes the book more complete as an advanced calculus text, and sections have been added introducing physical applications in thermodynamics, fluid dynamics, and classical rigid body mechanics.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a text written at the introductory level that dicusses the material leading up to, and including, Stoke’s theorem on manifolds. It is perhaps a little more elementary than Munkre’s excellent “Analysis on Manifolds” and considerably less-terse and much easier to read that Spivak’s “Calculus on manifolds”. Note though that while elementary, the material is not watered-down; the implicit/inverse function theorems, differential forms and other advanced material required for the statement and proof of Stokes is covered.One of the things I particularly like about the text is the author’s consistent and appropriate use of upper and lower indices to represent the components of contravariant and covariant vectors. The examples are usually well-chosen and help to clarify the material. The exercises are not difficult and tend to reinforce the material presented as opposed to developing addditional content and the answers to many of the computationally-oriented exercises are in the back of the book so that you can check your work. The book should be approachable by anyone with a good background in (rigorous) single-variable calculus and linear algebra.After assimilating the presented material, one should be well-prepared to venture in to more andvanced (and modern) texts on manifolds/differential geometry, e.g., as Lee’s Introduction to Smooth Manifolds

⭐I don’t recommend this book at all. The reason for three stars instead of a lower rating is because im open to the idea that I am the one who might not be mathematically mature enough for this book. First off, I want to say i utterly disagree with the review that said that this book is more elementary than munkres analysis on manifolds, Munkres book is very clearly written where as this book is plagued with confusing notation to the point where i found the text nearly incomprehensible. As an example in section 4.6 regarding the implicit function theorem the phi symbol is used to represent three different functions differing only in how bold the phi symbols are. There are very few examples in the main text and I don’t feel there are enough exercises to really begin to understand the subject. As alternatives, I highly recommend, hubbard and hubbard vector calculus linear algebra and differential forms, munkre’s text, James Callahan’s Advanced Calculus a geometric View, and Bucks Advanced calculus ( an incredibly well written book and my personal favorite that I’m surprised isn’t recommended by people when reddit threads,stack exchange posts are started asking for book recommendations come up). These books have very good exposition a wealth of exercises, (especially Buck and Callahan), ranging from medium to challenging to build up some confidence. I recommend any of those books over this one.

⭐Good text for independent study.

⭐In 1965 it was about all that was, but today we have many more choices to choose from. Perhaps if you are a genius you may still think this the best thing under the sun , but I am not quite there.The “advance calcuus” field has a ton of them now is what i am saying. And so does “smooth manifolds” theory. When I had “Calculus on Smooth Manifolds” by Spivak as the textbook from which I was learning, all this was obvious. That was a better middle ground between the two, even though it was inadequate in application examples.

⭐Aside from mathematical logic from Joe Ullian in my freshman year, this was my first real math course.It was taught in 1963, my sophomore year at the University of Chicago, by the topologist Eldon Dyer, from the preliminary edition at a time which felt like the dark ages in terms of calculus instruction. I remember the course and the book as amazing and inspiring. I was riveted by differential forms and Stokes Theorem and was ecstatic when Dyer, in a private conversation, gave a general description of de Rham cohomology (beyond the scope of the book).As mentioned by a previous reviewer, this book was a forerunner of what was to come and was about all that was available in resources of this kind at the time — the early 1960’s. But Fleming did us a wonderful favor in those days by pioneering (along with a few others) the presentation of calculus on manifolds at an early stage.

⭐Fleming gives a very solid, rigorous presentation of advanced calculus of several real variables. The implicit function theorem and inverse function theorem play central roles in the development of the theory. Fleming uses vector notation throughout, treating single variable calculus as a special case of the vector theory. Differential forms, exterior algebra, and manifolds are treated, as well as Lebesgue integration. Examples tend to focus on special cases and counter-examples. The book is a little light on practical applications, with the exception of the final chapter. I have only two substantial complaints with the book. First, the book often fails to build intuition about certain concepts. Second, there are relatively few problems devoted to computation, as applied mathematicians might desire. Strong points include the clarity of notation, rigor of proofs of theorems, and the treatment of both manifold theory and Lebesque integration. I strongly recommend this book, but caution that it may be slightly too advanced for all but the most serious undergraduate students. Working through this book will, however, build a level of mathematical maturity to handle more advanced analytical texts, such as Rudin.

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