Vector Calculus, 3rd Edition

(NOTE: Each chapter concludes with True/False Exercises and Miscellaneous Exercises.)

 

1. Vectors.

Vectors in Two and Three Dimensions. More About Vectors. The Dot Product. The Cross Product. Equations for Planes; Distance Problems. Some n-Dimensional Geometry. New Coordinate Systems.

 

2. Differentiation in Several Variables.

Functions of Several Variables; Graphing Surfaces. Limits. The Derivative. Properties; Higher-Order Partial Derivatives; Newton’s Method. The Chain Rule. Directional Derivatives and the Gradient.

 

3. Vector-Valued Functions.

Parametrized Curves and Kepler's Laws. Arclength and Differential Geometry. Vector Fields: An Introduction. Gradient, Divergence, Curl, and the Del Operator.

 

4. Maxima and Minima in Several Variables.

Differentials and Taylor's Theorem. Extrema of Functions. Lagrange Multipliers. Some Applications of Extrema.

 

5. Multiple Integration.

Introduction: Areas and Volumes. Double Integrals. Changing the Order of Integration. Triple Integrals. Change of Variables. Applications of Integration.

 

6. Line Integrals.

Scalar and Vector Line Integrals. Green's Theorem. Conservative Vector Fields.

 

7. Surface Integrals and Vector Analysis.

Parametrized Surfaces. Surface Integrals. Stokes's and Gauss's Theorems. Further Vector Analysis; Maxwell's Equations.

 

8. Vector Analysis in Higher Dimensions.

An Introduction to Differential Forms. Manifolds and Integrals of k-forms. The Generalized Stokes's Theorem.

 

Suggestions for Further Reading.

 

Answers to Selected Exercises.

 

Index.