Vector Calculus in Three Dimensions

In these notes we review the fundamentals of three-dimensional vector calculus. We will be surveying calculus on curves, surfaces and solid bodies in three-dimensional space. The three methods of integration — line, surface and volume (triple) integrals — and the fundamental vector differential operators — gradient, curl and divergence — are intimately related. The differential operators and integrals underlie the multivariate versions of the fundamental theorem of calculus, known as Stokes’ Theorem and the Divergence Theorem. A more detailed development can be found in any reasonable multi-variable calculus text, including [1, 6, 9].