Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds

✍ Scribed by Theodore Shifrin

• Publisher: John Wiley & Sons, Inc.
• Year: 2005
• Tongue: English
• Leaves: 504
• Edition: 1
• Category: Library

✦ Synopsis

Multivariable Mathematics combines linear algebra and multivariable calculus in a rigorous approach. The material is integrated to emphasize the role of linearity in all of calculus and the recurring theme of implicit versus explicit that persists in linear algebra and analysis. In the text, the author addresses 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 including complete proofs. By emphasizing the theoretical aspects and reviewing the linear algebra material quickly, the book can also be used as a text for an advanced calculus or multivariable analysis course culminating in a treatment of manifolds, differential forms, and the generalized Stokes’s Theorem.

✦ Table of Contents

Cover
Half-Title
Title Page
Copyright
Contents
Preface
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 Determinants and the Cross Product
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
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
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
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
6. Solving Nonlinear Problems
6.1 The Contraction Mapping Principle
6.2 The Inverse and Implicit Function Theorems
6.3 Manifolds Revisited
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
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

✦ Subjects

Linear Algebra, Rigorous Multivariable Calculus, Rigorous Calculus, Mathematical Analysis