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| Preface | |
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| Vectors and Matrices | |
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| Vectors in R n. 14 | |
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| Dot Product | |
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| Subspaces of R n | |
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| Linear Transformations and Matrix Algebra | |
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| Introduction to Determinates and the Cross Product | |
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| Functions, Limits, and Continuity | |
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| Scalar- and Vector-Valued Functions | |
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| A Bit of Topology in R n | |
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| Limits and Continuity | |
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| The Derivative | |
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| Partial Derivatives and Directional Derivatives | |
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| Differentiability | |
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| Differentiation Rules | |
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| The Gradient | |
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| Curves | |
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| Higher-Order Partial Derivatives | |
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| Implicit and Explicit Solutions of Linear Systems | |
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| Gaussian Elimination and the Theory of Linear Systems | |
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| Elementary Matrices and Calculating Inverse Matrices | |
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| Linear Independence, Basis, and Dimension | |
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| The Four Fundamental Subspaces | |
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| The Nonlinear Case: Introduction to Manifolds | |
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| Extremum Problems | |
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| Compactness and the Maximum Value Theorem | |
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| Maximum/Minimum Problems | |
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| Quadratic Forms and the Second Derivative Test | |
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| Lagrange Multipliers | |
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| Projections, Least Squares, and Inner Product Spaces | |
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| Solving Nonlinear Problems | |
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| The Contraction Mapping Principle | |
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| The Inverse and Implicit Function Theorems | |
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| Manifolds Revisited | |
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| Integration | |
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| Multiple Integrals | |
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| Iterated Integrals and Fubini?s Theorem | |
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| Polar, Cylindrical, and Spherical Coordinates | |
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| Physical Applications | |
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| Determinants and n-Dimensional Volume | |
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| Change of Variables Theorem | |
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| Differential Forms and Integration on Manifolds | |
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| Motivation | |
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| Differential Forms | |
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| Line Integrals and Green?s Theorem | |
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| Surface Integrals and Flux | |
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| Stokes?s Theorem | |
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| Applications to Physics | |
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| Applications to Topology | |
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| Eigenvalues, Eigenvectors, and Applications | |
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| Linear Transformations and Change of Basis | |
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| Eigenvalues, Eigenvectors, and Diagonalizability | |
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| Difference Equations and Ordinary Differential Equations | |
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| The Spectral Theorem | |
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| Glossary of Notations and Results from Single-Variable Calculus | |
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| For Further Reading | |
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| Answers to Selected Exercises | |
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| Index | |