Multivariable Mathematics / Edition 1

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Overview

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.

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Product Details

  • ISBN-13: 9780471526384
  • Publisher: Wiley
  • Publication date: 1/23/2004
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 504
  • Sales rank: 937,062
  • Product dimensions: 7.72 (w) x 9.39 (h) x 0.95 (d)

Table of Contents

Preface.

Chapter 1. Vectors and Matrices.

1.1 Vectors in Rn..

1.2 Dot Product.

1.3 Subspaces of Rn.

1.4 Linear Transformations and Matrix Algebra.

1.5 Introduction to Determinates and the Cross Product.

Chapter 2. Functions, Limits, and Continuity.

2.1. Scalar- and Vector-Valued Functions.

2.2. A Bit of Topology in Rn.

2.3. Limits and Continuity.

Chapter 3. The Derivative.

3.1. Partial Derivatives and Directional Derivatives.

3.2. Differentiability.

3.3. Differentiation Rules.

3.4. The Gradient.

3.5. Curves.

3.6. Higher-Order Partial Derivatives.

Chapter  4. Implicit and Explicit Solutions of Linear Systems.

4.1. Gaussian Elimination and the Theory of Linear Systems.

4.2. Elementary Matrices and Calculating Inverse Matrices.

4.3. Linear Independence, Basis, and Dimension.

4.4. The Four Fundamental Subspaces.

4.5. The Nonlinear Case: Introduction to Manifolds.

Chapter 5. Extremum Problems.

5.1. Compactness and the Maximum Value Theorem.

5.2. Maximum/Minimum Problems.

5.3. Quadratic Forms and the Second Derivative Test.

5.4. Lagrange Multipliers.

5.5. Projections, Least Squares, and Inner Product Spaces. 

Chapter 6. Solving Nonlinear Problems.

6.1. The Contraction Mapping Principle.

6.2. The Inverse and Implicit Function Theorems.

6.3. Manifolds Revisited.

Chapter 7. Integration.

7.1. Multiple Integrals.

7.2. Iterated Integrals and Fubini’s Theorem.

7.3. Polar, Cylindrical, and Spherical Coordinates.

7.4. Physical Applications.

7.5. Determinants and n-Dimensional Volume.

7.6. Change of Variables Theorem.

Chapter 8. Differential Forms and Integration on Manifolds.

8.1. Motivation.

8.2. Differential Forms.

8.3. Line Integrals and Green’s Theorem.

8.4. Surface Integrals and Flux.

8.5. Stokes’s Theorem.

8.6. Applications to Physics.

8.7. Applications to Topology.

9.  Eigenvalues, Eigenvectors, and Applications.

9.1. Linear Transformations and Change of Basis.

9.2. Eigenvalues, Eigenvectors, and Diagonalizability.

9.3. Difference Equations and Ordinary Differential Equations.

9.4. The Spectral Theorem.

Glossary of Notations and Results from Single-Variable Calculus.

For Further Reading.

Answers to Selected Exercises.

Index.

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  • Anonymous

    Posted August 8, 2004

    highly recommended

    This book does a very clear and scholarly job of combining linear algebra and higher dimensional calculus. The exposition is excellent, and the mathematics is rigorous without being overly technical. I have only browsed it, but I especially liked chapter 8, where several beautiful and useful versions of the stokes theorem are explained, proved and applied, both to physics, and to some very non trivial results from topology. I recommend this as a place for a good student with a solid background in one variable calculus to learn both linear algebra, and its application to the most natural modern formulation of several variable calculus. There are also lots of good problems, presented with wit and humor (as in one example taken from car talk). If you read cyrillic you will also recognize a tribute to another outstanding text in this genre.

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