Integration in several variables begins with area and volume, but it quickly grows into a broader theory of integration over regions, curves, and surfaces. To handle these domains effectively, we need both good coordinates and a clear treatment of orientation.
We start with multiple integrals and change of variables, then introduce vector fields, differential forms, and pullbacks. With this language in place, we define line and surface integrals and arrive at the fundamental theorems of vector calculus as special cases of general Stokes’ theorem.
