We introduce Stokes’ theorem.

Our final fundamental theorem of calculus is Stokes’ theorem. Historically speaking, Stokes’ theorem was discovered after both Green’s theorem and the divergence theorem. Its application is probably the most obscure, with the primary applications being rooted in electricity-and-magnetism and fluid dynamics.

Nevertheless, once we know Stokes’ theorem, we can view it as a direct generalization of Green’s theorem.

If we compare the conclusion of Stokes’ theorem to the conclusion of Green’s theorem, we see that they are quite similar:

\[ \underbrace {\iint _R \curl \vec {F}\d A = \oint _{\partial R} \vec {F}\dotp \d \vec {p}}_{\text {Green's Theorem}}\qquad \underbrace {\iint _R (\curl \vec {F})\dotp \uvec {n}\d S = \oint _{\partial R} \vec {F}\dotp \d \vec {p}}_{\text {Stokes' Theorem}} \]

Like Green’s theorem, Stokes’ theorem computes circulation along a surface:

In essence, Green’s theorem is nothing more than Stokes’ theorem when we assume that the surface is restricted to the \((x,y)\)-plane.

1 Transforming integrals

Like the divergence theorem, one application of Stokes’ theorem is to transform a difficult integral into an easier one.

2 Swapping surfaces

Since Stokes’ theorem says that if we want to evaluate the circulation along a surface, we need only look at the boundary, and we can sometimes be very clever and swap one surface for another, provided that they have the same boundary! Consider the pictures below:

Above we see a region \(R\) on the left and a region \(D\) on the right. In the middle is the boundary of both regions. Since both regions \(R\) and \(D\) have the same boundary, we can effectively swap surfaces by using Stokes’ Theorem. Check out the next example.

3 Our final fundamental theorem of calculus

How is Stokes’ Theorem a fundamental theorem of calculus? Well consider this:

Are there no more fundamental theorems of calculus? Well, there are, but as you will see if you continue your study of mathematics, they are all generalizations of the theorems we already know.

For some interesting extra reading check out: