Advanced Calculus of Several Variables (Dover Books on Mathematics) by C. H. Edwards Jr. (PDF)

Description

In this high-level treatment, the author provides a modern conceptual approach to multivariable calculus, emphasizing the interplay of geometry and analysis via linear algebra and the approximation of nonlinear mappings by linear ones. At the same time, the book gives equal attention to the classical applications and computational methods responsible for much of the interest and importance of this subject.Beginning with a discussion of Euclidean space and linear mappings, Professor Edwards (University of Georgia) follows with a thorough and detailed exposition of multivariable differential and integral calculus. Among the topics covered are the basics of single-variable differential calculus generalized to higher dimensions, the use of approximation methods to treat the fundamental existence theorems of multivariable calculus, iterated integrals and change of variable, improper multiple integrals and a comprehensive discussion, from the viewpoint of differential forms, of the classical material associated with line and surface integrals, Stokes’ theorem, and vector analysis. The author closes with a modern treatment of some venerable problems of the calculus of variations.Intended for students who have completed a standard introductory calculus sequence, the book includes many hundreds of carefully chosen examples, problems, and figures. Indeed, the author has devoted a great deal of attention to the 430 problems, mainly concrete computational ones, that will reward students who solve them with a rich intuitive and conceptual grasp of the material.

User’s Reviews

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

⭐I started reading this book in August or late July ’21. I finished in December except skipping the Calculus of Variations chapter.It was easy enough to read, but understanding so you can attempt to do exercises is the tough slog. The material flows together as if once you finally get to Lagrange Multipliers, quadratic forms, and Taylor series the progression makes sense. Then we start on integration and it seems dated and uses unnecessary notions, at least unnecessary to me, bounded support, for example. I finished off with Stokes’ Theorem. This was for a course of Advanced Calculus at Uni.The problems can be insane, but they are also easy. I barely supplemented the book with Bachman’s. My book is tattered after the semester.

⭐This book can be used to supplement existing texts used for standard Advanced Calculus courses, and it includes all the essentials one would look for such as:Linear mappings (Ch. I), multivariable calculus (Ch. II), Multiple Integrals (Ch. IV), Line and suurface integrals (Ch. V) and calculus of variations (Ch. VI).My only quibble is the approach is somewhat dated, and also I thought it would have been nice to have at least one chapter on complex calculus – treating contour integration, Laurent series, and the residue theorem and applications to complex integration. Most Advanced Calc courses today include such fare, but again – if Edwards’ text is just used as a supplement it’s no biggie.The text is also very useful for self-study and if the student wants an additional text to cover complex calculus he can get ‘Complex Variables and Applications’ by James Ward Brown and Ruel V. Churchill (6th ed.)

⭐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.

⭐So far, this text is an invaluable reference for me.It has the more theoretical bent that I desired, and is very reader-friendly in the following ways:(1) The book is compact yet includes full proofs, no-nonsense explanations, and interesting examples.(2) The organization is consistent—specific topic are easily found from the index or table of contents, and there is balanced depth from part to part.(3) The exercises are accessible, although there is no solution appendix, and elucidate the discussion.In short, this book would be helpful for anyone who is interested in theoretical mathematics, such as the student of Physics or Mathematics, who has a background of single-variable Calculus.Note: The newer /Multivariable Mathematics/ by Shifrin covers content similar to that of this book.However, the former is more wordy and self-conscious, which makes is more difficult to read.

⭐Great information. Interesting applications mentioned. Sometimes it felt a little convoluted by the notation/wordings, but I learned a lot reading it.

⭐Not abstract as the Edwards book, but competently written for an engineering/physics student.

⭐I actually don’t believe that there are a lot of typos in this textbook when one of my classmates complained about it during the first class. When I read it for the first ten pages, I discovered a couple of typos which really makes me difficult to understand…. Really upset that the author never updated those typos (I am definitely sure that previous readers have sent those typos…. – -|||)

⭐I am currently using this book to self teach myself multivariable calculus. The writing style is clear enough, and the gap filling is (in my opinion) something kind of nice. However, what makes this book difficult to use are some glaring errors, especially in the problems. For example, problem 2.5(d) is (and confirmed by Wolfram Alpha) impossible- the partial derivative at x for the given function IS continuous at (0,0).

⭐One of the challenges in progressing through high levels of math is that some of the core material — calculus and linear algebra — may have been treated as a more superficial level than needed to understand more advanced courses. This book is the perfect remedy, treating multivariable calculus at the perfect level of rigor. Spivak’s “Calculus on Manifolds” plays a similar role, but I like this alternative better, it takes things a little more slowly and includes useful exercises.

⭐The content is very abstract and explanations are frequently implied instead of a step by step approach. although rigorous, it isn’t self- teaching friendly

⭐Gostei da profundidade como trata o tema.This is in fact a beautiful text on the calculus of functions of several variables. It assumes some familarity with calculus in 1-d but not too much. Edwards is a very well known teacher and this book breathes his classroom experience. It moves from motivations and heuristics to rigorous theory and gives plenty of non-trivial examples.

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