Vector Calculus

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
   • True/False Exercises for Chapter 1
   • Miscellaneous Exercises for Chapter 1
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)
   • True/False Exercises for Chapter 2
   • Miscellaneous Exercises for Chapter 2
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
   • True/False Exercises for Chapter 3
   • Miscellaneous Exercises for Chapter 3
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
   • True/False Exercises for Chapter 4
   • Miscellaneous Exercises for Chapter 4
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)
   • True/False Exercises for Chapter 5
   • Miscellaneous Exercises for Chapter 5
6. Line Integrals
   • 6.1 Scalar and Vector Line Integrals
   • 6.2 Green's Theorem
   • 6.3 Conservative Vector Fields
   • True/False Exercises for Chapter 6
   • Miscellaneous Exercises for Chapter 6
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
   • True/False Exercises for Chapter 7
   • Miscellaneous Exercises for Chapter 7
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
   • True/False Exercises for Chapter 8
   • Miscellaneous Exercises for Chapter 8

Suggestions for Further Reading

Answers to Selected Exercises

Index

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