Textbook
Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and ManifoldsJanuary 2004, ©2005

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 VectorValued 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. HigherOrder 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 nDimensional 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 SingleVariable Calculus.
For Further Reading.
Answers to Selected Exercises.
Index.
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