An introduction to multivariable mathematics
By: Leon Simon
Series: Synthesis lectures on mathematics and statistics -- #3
Publisher: Morgan & Claypool Publishers
Published: 2008
Place of publication: San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA)
Language: eng
Format: [electronic resource] /
ISBN-13: 9781598298024; 9781598298017
Subjects: Linear Algebras; Mathematical analysis
Other titles: Synthesis digital library of engineering and computer science.
Description: The text is designed for use in a 40 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 10 or 11 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 9 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 4 lecture per week program in the Autumn Quarter each year, the first 6 lectures of the 9 lecture appendix are presented at the rate of one lecture a week in weeks 2-7 of the quarter, with the remaining 3 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 1 quarter or 1 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.
Notes: Part of: Synthesis digital library of engineering and computer science.

Title from PDF t.p. (viewed on October 15, 2008).

Series from website.

Includes index.

Abstract freely available; full-text restricted to subscribers or individual document purchasers.

Also available in print.

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader.
Table of contents: title: Linear algebra; title: Vectors in Rn; title: Dot product and angle between vectors in Rn; title: Subspaces and linear dependence of vectors; title: Gaussian elimination and the linear dependence lemma; title: The basis theorem; title: Matrices; title: Rank and the rank-nullity theorem; title: Orthogonal complements and orthogonal projection; title: Row echelon form of a matrix; title: Inhomogeneous systems; title: Analysis in Rn; title: Open and closed sets in Euclidean space; title: Bolzano-Weierstrass, limits and continuity in Rn; title: Differentiability; title: Directional derivatives, partial derivatives, and gradient; title: Chain rule; title: Higher-order partial derivatives; title: Second derivative test for extrema of multivariable function; title: Curves in Rn; title: Submanifolds of Rn and tangential gradients; title: More linear algebra; title: Permutations; title: Determinants; title: Inverse of a square matrix; title: Computing the inverse; title: Orthonormal basis and Gram-Schmidt; title: Matrix representations of linear transformations; title: Eigenvalues and the spectral theorem; title: More analysis in Rn; title: Contraction mapping principle; title: Inverse function theorem; title: Implicit function theorem; title: Introductory lectures on real analysis; title: Lecture 1: The real numbers; title: Lecture 2: Sequences of real numbers and the Bolzano-Weierstrass theorem; title: Lecture 3: Continuous functions; title: Lecture 4: Series of real numbers; title: Lecture 5: Power series; title: Lecture 6: Taylor series representations; title: Lecture 7: Complex series, products of series, and complex exponential series; title: Lecture 8: Fourier series; title: Lecture 9: Pointwise convergence of trigonometric Fourier series.