Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.
Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by Hubbard and Hubbard. This text includes proofs of the major theorems of vector calculus and, as a great benefit to the self-learner, a solutions manual for many of the problems so that you can check your work. It should be approachable by anyone with a good background in basic calculus.
Solutions Manual Elementary Analysis The Theory Of Calculus
Wendell Fleming's Functions of Several Variables is a somewhat more sophisticated treatment of the subject but still elementary and approachable. The author works up to proving Stoke's Theorem on manifolds. It does include measure theory but really only enough for integration in a general context to make sense.
My personal advise is the two volumes by Zorich - "Mathematical Analysis vol. 1" and "Mathematical Analysis vol. 2". Take a careful look at the table of contents of both since they deal with all rigorous calculus needed from real numbers and functions of one variable to multivariable calculus and vector analysis, curves and surfaces, differential forms, series and asymptotic methods. Most proofs are included: from the usual easy rules and techniques of differentiation and indefinite integration in one variable up to very important results in multivariable calculus like the general Stoke's theorem and the change of variables formula inside of a multiple Riemann integral (a fundamental result not proved in many other books as rigorously beyond heuristic justifications for double or triple integrals).
I really think that they are a wonderful complete reference for the real analysis needed just before entering general measure theory (Lebesgue integration) and other more advanced topics such as operators and Hilbert spaces (as a great complement to Zorich, dealing with this adv. subjects a great book is Kantorovitz - "Introduction to Modern Analysis")
I have lecture notes on multivariable calculus and Stokes's theorem on my web page ( ). They are free. There are solutions in the back to many of the problems. The coverage of the basic material up to manifolds is fairly comprehensive.
This set includes: Analysis in Vector Spaces ISBN 978-0-470-14824-2 and Analysis in Vector Spaces, Student Solutions Manual ISBN 978-0-470-14825-9.rigorous introduction to calculus in vector spacesThe concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences.The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes:* Sets and functions* Real numbers* Vector functions* Normed vector spaces* First- and higher-order derivatives* Diffeomorphisms and manifolds* Multiple integrals* Integration on manifolds* Stokes' theorem* Basic point set topologyNumerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants.Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.
Is the embedding tag really useful? There are way too many things that can fall under this tag. From elementary embeddings in set theory, in model theory, to embedding of topological spaces, and to many other fields.
Currently there are about 20 questions carrying either tag. Neither has a tag wiki nor a follower. The topics of the questions with the lower-bound tag range from number theory to real analysis to random graphs to quadratic programming.
This is a text on elementary multivariable calculus, designed for students who have completed courses in single-variable calculus. The traditional topics are covered: basic vector algebra; lines, planes and surfaces; vector-valued functions; functions of 2 or 3 variables; partial derivatives; optimization; multiple integrals; line and surface integrals. 2ff7e9595c
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