Vector Calculus

Susan Colley is the Andrew and Pauline Delaney Professor of Mathematics at Oberlin College and currently Chair of the Department, having also previously served as Chair. She received S.B. and Ph.D. degrees in mathematics from the Massachusetts Institute of Technology prior to joining the faculty at Oberlin in 1983. Her research focuses on enumerative problems in algebraic geometry, particularly concerning multiple-point singularities and higher-order contact of plane curves. Professor Colley has published papers on algebraic geometry and commutative algebra, as well as articles on other mathematical subjects. She has lectured internationally on her research and has taught a wide range of subjects in undergraduate mathematics. Professor Colley is a member of several professional and honorary societies, including the American Mathematical Society, the Mathematical Association of America, Phi Beta Kappa, and Sigma Xi.

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