We introduce Stokes’ theorem.

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

### Transforming integrals

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

### 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 on the left and a region on the right. In the middle is the
boundary of *both* regions. Since both regions and have the same boundary, we
can effectively *swap* surfaces by using Stokes’ Theorem. Check out the next
example.

### 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: