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An introduction to multivariable mathematics
Author
Title
An introduction to multivariable mathematics [electronic resource] / Leon Simon.
ISBN
9781598298024 (electronic bk.)
9781598298017 (pbk.)
Published
San Rafael, Calif. (1537 Fourth Street, San Rafael, CA 94901 USA) : Morgan & Claypool Publishers, c2008.
Physical Description
1 electronic text (x, 132 p. : ill.) : digital file.
Local Notes
Access is available to the Yale community.
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.
Access and use
Access is restricted by licensing agreement.
Summary
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.
Variant and related titles
Synthesis digital library of engineering and computer science.
Other formats
Also available in print.
Format
Books / Online
Language
English
Added to Catalog
January 19, 2011
Series
Synthesis lectures on mathematics and statistics ; #3
System details note
Mode of access: World Wide Web.
System requirements: Adobe Acrobat Reader.
Contents
Linear algebra
Vectors in Rn
Dot product and angle between vectors in Rn
Subspaces and linear dependence of vectors
Gaussian elimination and the linear dependence lemma
The basis theorem
Matrices
Rank and the rank-nullity theorem
Orthogonal complements and orthogonal projection
Row echelon form of a matrix
Inhomogeneous systems
Analysis in Rn
Open and closed sets in Euclidean space
Bolzano-Weierstrass, limits and continuity in Rn
Differentiability
Directional derivatives, partial derivatives, and gradient
Chain rule
Higher-order partial derivatives
Second derivative test for extrema of multivariable function
Curves in Rn
Submanifolds of Rn and tangential gradients
More linear algebra
Permutations
Determinants
Inverse of a square matrix
Computing the inverse
Orthonormal basis and Gram-Schmidt
Matrix representations of linear transformations
Eigenvalues and the spectral theorem
More analysis in Rn
Contraction mapping principle
Inverse function theorem
Implicit function theorem
Introductory lectures on real analysis
Lecture 1: The real numbers
Lecture 2: Sequences of real numbers and the Bolzano-Weierstrass theorem
Lecture 3: Continuous functions
Lecture 4: Series of real numbers
Lecture 5: Power series
Lecture 6: Taylor series representations
Lecture 7: Complex series, products of series, and complex exponential series
Lecture 8: Fourier series
Lecture 9: Pointwise convergence of trigonometric Fourier series.
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