An Introduction to Multivariable Mathematics (Synthesis Lectures on Mathematics and Statistics, 3) 0th Edition by Leon Simon | (PDF) Free Download

Description

The text is designed for use in a forty-lecture introductory course covering linear algebra, multivariable differential calculus, and an introduction to real analysis. The core material of the book is arranged to allow for the main introductory material on linear algebra, including basic vector space theory in Euclidean space and the initial theory of matrices and linear systems, to be covered in the first ten or eleven lectures, followed by a similar number of lectures on basic multivariable analysis, including first theorems on differentiable functions on domains in Euclidean space and a brief introduction to submanifolds. The book then concludes with further essential linear algebra, including the theory of determinants, eigenvalues, and the spectral theorem for real symmetric matrices, and further multivariable analysis, including the contraction mapping principle and the inverse and implicit function theorems. There is also an appendix which provides a nine-lecture introduction to real analysis. There are various ways in which the additional material in the appendix could be integrated into a course–for example in the Stanford Mathematics honors program, run as a four-lecture per week program in the Autumn Quarter each year, the first six lectures of the nine-lecture appendix are presented at the rate of one lecture per week in weeks two through seven of the quarter, with the remaining three lectures per week during those weeks being devoted to the main chapters of the text. It is hoped that the text would be suitable for a quarter or semester course for students who have scored well in the BC Calculus advanced placement examination (or equivalent), particularly those who are considering a possible major in mathematics. The author has attempted to make the presentation rigorous and complete, with the clarity and simplicity needed to make it accessible to an appropriately large group of students. Table of Contents: Linear Algebra / Analysis in R / More Linear Algebra / More Analysis in R / Appendix: Introductory Lectures on Real Analysis

User’s Reviews

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

⭐First of all, let me make it clear that this is not for regular college students. This book was written with MATH 51H (Honors Multivariable Mathematics) in mind, which Prof. Simon teaches every fall at Stanford University. As an honors course for math majors, it is entirely proof-based, relying on basic axioms and definitions of concepts and building its way up through theorems.The first chapter introduces basic linalg concepts like vectors, subspaces, matrices, and row reduction. The second chapter includes open/closed sets, continuity, differentiability, partial derivatives, and submanifolds. The third chapter focuses on matrices, with topics like determinants and eigenvalues. The fourth chapter presents some more advanced theorems in analysis. Finally, the appendix contains nine lectures in analysis, beginning with the definition of real numbers, exploring sequences/series and Bolzano-Weierstrass, and then going into Taylor series.The book is a pleasant but slow read and contains many thought-provoking exercises. Simon is very clear and rigorous throughout the presentation of the material. I highly recommend it (and if you’re in his class, you have no choice but to get it).

⭐I learned the introductory real analysis and linear algebra in a class taught by Leon Simon himself. Prof. Simon is one of the best lecturers at Stanford and this textbook is the best proof for this. I especially like the way he arranges the materials, which reveals a deep connection between linear algebra and multivariable calculus. I think Stanford will keep using this book for a long time.

⭐I mostly agree with Tony J.’s review above- Simon’s course is wonderfully clear and insightful and his choice of material is excellent. The exercises are challenging but not too difficult for talented students. I particularly liked how Simon interwove tightly the theory of linear algebra and the differentiability of functions in Euclidean space. But there’s a huge problem here and it has to do with what’s NOT in the book. A theory of integration is completely avoided here-and the end result is that the only part of multivariable analysis that gets covered here is the easy part-the part dealing with differentiation and it’s relationship to linear algebra. There’s also nothing on multi-linear algebra and differential forms-again, this is because all aspects of integration theory is avoided, which generally is how these topics are motivated in such a course.It’s quite probable that Professor Simon only taught the first half of the course at Stanford-and that’s fine. But unfortunately, that greatly limits the use of this book as a general textbook on rigorous n-dimensional calculus/real analysis. It will be useful for the first half of such a course, but then the instructor and students are going to have to make use of one of the standard sources for these courses, such as Munkres’ ANALYSIS ON MANIFOLDS or Fleming’s FUNCTIONS OF SEVERAL VARIABLES. In other words, a good book with very limited range.

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