Multivariable Integration

The one-dimensional integral sums a function along a line, and the fundamental theorem of calculus tells you how to undo that sum. Multivariable integration asks the same question of a surface, a solid, and a curve, and the answers turn out to be considerably richer: a double integral adds up values over a region of the plane, a triple integral does the same over a solid in space, a line integral accumulates along a curve, and a surface integral measures flow through a surface. The surprising part is that these four apparently different constructions are welded together by three theorems — Green’s, Gauss’, and Stokes’ — which turn otherwise unwieldy computations into boundary calculations you can do in one line.

This course works through the concepts, the computation rules, and the applications in that order, and it is written for readers who already know single-variable integration and partial derivatives. By the end you should be able to set up a double or triple integral in rectangular, polar, cylindrical, and spherical coordinates, compute line and surface integrals of both kinds, and recognise when a flux or circulation integral is the right tool for the problem in front of you.