In this section, we study Stokes’ theorem, a higher-dimensional generalization of Green’s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green’s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions. Stokes’ theorem relates a vector surface integral over surface [latex]S[/latex] in space to a line integral around the boundary of [latex]S[/latex]. Therefore, just as the theorems before it, Stokes’ theorem can be used to reduce an integral over a geometric object [latex]S[/latex] to an integral over the boundary of [latex]S[/latex].

In addition to allowing us to translate between line integrals and surface integrals, Stokes’ theorem connects the concepts of curl and circulation. Furthermore, the theorem has applications in fluid mechanics and electromagnetism. We use Stokes’ theorem to derive Faraday’s law, an important result involving electric fields.