Table of Contents:

1. Vectors

• 1.1 Vectors in Two and Three Dimensions

• 1.2 More About Vectors

• 1.3 The Dot Product

• 1.4 The Cross Product

• 1.5 Equations for Planes; Distance Problems

• 1.6 Some n-dimensional Geometry

• 1.7 New Coordinate Systems

2. Differentiation in Several Variables

• 2.1 Functions of Several Variables;Graphing Surfaces

• 2.2 Limits

• 2.3 The Derivative

• 2.4 Properties; Higher-order Partial Derivatives

• 2.5 The Chain Rule

• 2.6 Directional Derivatives and the Gradient

• 2.7 Newton's Method (optional)

3. Vector-Valued Functions

• 3.1 Parametrized Curves and Kepler's Laws

• 3.2 Arclength and Differential Geometry

• 3.3 Vector Fields: An Introduction

• 3.4 Gradient, Divergence, Curl, and the Del Operator

4. Maxima and Minima in Several Variables

• 4.1 Differentials and Taylor's Theorem

• 4.2 Extrema of Functions

• 4.3 Lagrange Multipliers

• 4.4 Some Applications of Extrema

5. Multiple Integration

• 5.1 Introduction: Areas and Volumes

• 5.2 Double Integrals

• 5.3 Changing the Order of Integration

• 5.4 Triple Integrals

• 5.5 Change of Variables

• 5.6 Applications of Integration

• 5.7 Numerical Approximations of Multiple Integrals (optional)

6. Line Integrals

• 6.1 Scalar and Vector Line Integrals

• 6.2 Green's Theorem

• 6.3 Conservative Vector Fields

7. Surface Integrals and Vector Analysis

• 7.1 Parametrized Surfaces

• 7.2 Surface Integrals

• 7.3 Stokes's and Gauss's Theorems

• 7.4 Further Vector Analysis; Maxwell's Equations

8. Vector Analysis in Higher Dimensions

• 8.1 An Introduction to Differential Forms

• 8.2 Manifolds and Integrals of k-forms

• 8.3 The Generalized Stokes's Theorem

Suggestions for Further Reading

Answers to Selected Exercises

Index

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