Chapter 5: Multiple Integrals and Vector Calculus

Integration in the plane starts where integration on the line did — chop the region into cells, weight each cell by a function value, add, refine — but the cells now have shape, and shape is where the interesting mathematics lives. Chop a disk along rings and rays and the cells are not congruent: they widen in proportion to their distance from the center. That widening, made visible on a mesh you can refine at will, is the whole truth behind the polar area element $rdrd\theta$ — the mysterious extra $r$ is something you can count, cell by cell, before any formula asserts it.

The same idea scales to every change of variables. A substitution is a map that carries a square grid to a curvilinear mesh, and the Jacobian determinant is nothing but the local area-scaling of its cells — the polar $r$ is one Jacobian among many. Watching a unit square shear, bend, and locally shrink under a two-slider map turns the scariest-looking theorem of multivariable integration into a statement about how mesh cells change size.

Then the integrands themselves become geometric: vector fields. A field assigns an arrow to every point, and the fundamental question — is it the gradient of some potential function, or not? — turns out to be answerable with a single closed loop. A field whose arrows cross the loop everywhere does no net work around it; a field whose arrows circulate along the loop cannot be a gradient, and one honest line integral proves it.

The chapter, and the book, close with Stokes' Theorem: circulation around a boundary curve equals the flux of curl through any cap spanning it. Two very different surfaces on one shared rim, threaded by the same swirling field, force the theorem's strangest consequence — the cap is disposable, and only the boundary owns the answer.