Functions of Several Variables (Undergraduate Texts in Mathematics)

Rating:
Summary: Very solid text
Review: 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.

Rating:
Summary: Very solid text
Review: 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.

Rating:
Summary: Well presented
Review: This book revolves around three theorems: the inverse function theorem, the implicit function theorem, and Stokes' theorem. The prerequisites are a working knowledge of Linear algebra and undergraduate calculus. What distinguishes this text from other books on advanced calculus is that it focuses at the outset on R^N instead of on the real line. The advantage of starting out with R^N is that the reader becomes more quickly accustomed to the notation and can subsequently interpret the real line is a special case more easily. Later sections in the book cover exterior algebra and differential calculus and integration on manifolds. There is a discussion of the Lebesgue integral in R^N also. The notation is very clean and there are interesting exercises with corresponding numerical answers in the appendix. The writing is well paced, with a uniform level of difficulty throughout. I recommend this for an advanced undergraduate or beginning graduate student.