Multivariable Mathematics
Linear Algebra, Multivariable Calculus, and Manifolds
1. Auflage Januar 2004
512 Seiten, Hardcover
Wiley & Sons Ltd
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.
Chapter 1. Vectors and Matrices.
1.1 Vectors in R¯n..
1.2 Dot Product.
1.3 Subspaces of R¯n.
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 R¯n.
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.
