Vector Calculus

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

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