Multivariable Mathematics Linear Algebra, Multivariable Calculus, and Manifolds

ISBN-10: 047152638X
ISBN-13: 9780471526384
Edition: 2005
Authors: Theodore Shifrin
List price: $213.95
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Description: Multivariable Mathematics combines linear algebra and multivariable mathematics in a rigorous approach. The material is integrated to emphasize the recurring theme of implicit versus explicit that persists in linear algebra and analysis. In the  More...

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Book details

List price: $213.95
Copyright year: 2005
Publisher: John Wiley & Sons, Incorporated
Publication date: 1/26/2004
Binding: Hardcover
Pages: 504
Size: 7.50" wide x 9.25" long x 1.00" tall
Weight: 2.486
Language: English

Multivariable Mathematics combines linear algebra and multivariable mathematics in a rigorous approach. The material is integrated to emphasize the recurring theme of implicit versus explicit that persists in linear algebra and analysis. In the text, the author includes all of the standard computational material found in the usual linear algebra and multivariable calculus courses, and more, interweaving the material as effectively as possible, and also includes complete proofs. * Contains plenty of examples, clear proofs, and significant motivation for the crucial concepts. * Numerous exercises of varying levels of difficulty, both computational and more proof-oriented. * Exercises are arranged in order of increasing difficulty.

Preface
Vectors and Matrices
Vectors in R n. 14
Dot Product
Subspaces of R n
Linear Transformations and Matrix Algebra
Introduction to Determinates and the Cross Product
Functions, Limits, and Continuity
Scalar- and Vector-Valued Functions
A Bit of Topology in R n
Limits and Continuity
The Derivative
Partial Derivatives and Directional Derivatives
Differentiability
Differentiation Rules
The Gradient
Curves
Higher-Order Partial Derivatives
Implicit and Explicit Solutions of Linear Systems
Gaussian Elimination and the Theory of Linear Systems
Elementary Matrices and Calculating Inverse Matrices
Linear Independence, Basis, and Dimension
The Four Fundamental Subspaces
The Nonlinear Case: Introduction to Manifolds
Extremum Problems
Compactness and the Maximum Value Theorem
Maximum/Minimum Problems
Quadratic Forms and the Second Derivative Test
Lagrange Multipliers
Projections, Least Squares, and Inner Product Spaces
Solving Nonlinear Problems
The Contraction Mapping Principle
The Inverse and Implicit Function Theorems
Manifolds Revisited
Integration
Multiple Integrals
Iterated Integrals and Fubini?s Theorem
Polar, Cylindrical, and Spherical Coordinates
Physical Applications
Determinants and n-Dimensional Volume
Change of Variables Theorem
Differential Forms and Integration on Manifolds
Motivation
Differential Forms
Line Integrals and Green?s Theorem
Surface Integrals and Flux
Stokes?s Theorem
Applications to Physics
Applications to Topology
Eigenvalues, Eigenvectors, and Applications
Linear Transformations and Change of Basis
Eigenvalues, Eigenvectors, and Diagonalizability
Difference Equations and Ordinary Differential Equations
The Spectral Theorem
Glossary of Notations and Results from Single-Variable Calculus
For Further Reading
Answers to Selected Exercises
Index

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