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.
Chapter 2. Functions, Limits, and Continuity.
Chapter 3. The Derivative.
Chapter 4. Implicit and Explicit Solutions of Linear Systems.
Chapter 5. Extremum Problems.
Chapter 6. Solving Nonlinear Problems.
Chapter 7. Integration.
Chapter 8. Differential Forms and Integration on Manifolds.
Chapter 9. Eigenvalues, Eigenvectors, and Applications.
Glossary of Notations and Results from Single-Variable Calculus.
For Further Reading.
Answers to Selected Exercises.