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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
Nevertheless, once we know Stokes’ theorem, we can view it as a direct generalization
of Green’s theorem.
Stokes’ Theorem If the components of have continuous partial derivatives and is a
boundary of a closed surface and parameterizes in a counterclockwise direction
with the interior on the left, then:
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:
In essence, Green’s theorem is nothing more than Stokes’ theorem when we assume
that the surface is restricted to the -plane.
Like the divergence theorem, one application of Stokes’ theorem is to transform a
difficult integral into an easier one.
Consider the line integral where is parameterized by , . Compute this integral
using Stokes’ theorem.
Let be the surface bounded by , so in this case .
Moreover, set Now we may write: and Stokes’ theorem says: Write with me,
Hence but Hence
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
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
Consider the surface integral where and is the upper hemisphere of the sphere:
Compute this integral using Stokes’ theorem.
Note that the boundary of the sphere is
identical to the boundary of the disk where . Hence we may dispense with our
“upper hemisphere” and simply work with the disk of radius . Write with me, Hence
By Stokes’ theorem this is equal to the integral we desire.
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.