Section 16.8: Stoke's Theorem
Goals:
1. To use Stoke's Theorem to evaluate a line integral
2. To use curl to analyze the motion of a rotating liquid
Stokes Theorem relates line integrals of vector fields to surface integrals of vector fields.
Stokes Theorem
Let S be an oriented piecewise-smooth surface (with unit normal vector N) that is bounded
by a simple, closed, piecewise-smooth boundary curve C with positive orientation. Let F be
a vector field whose components have continuous partial derivatives on an open space
region that contains S. Then
∫ F • dr = ∫∫ (curl F) • NdS
C
Notes:
S
1. Think of the space curve C as the rim of the surface (see picture below).
2. For consistency, a positive orientation of C means that if you were to walk along C with
your head pointed towards N, then the surface will always be on your left.
3. Stokes theorem implies that we can get the same value for any two surface integrals as
long as both surfaces have the same rim, C.
4. Stokes theorem can help us interpret the curl of a vector field, as described below.
Curl and Circulation
Just as the divergence theorem will assist us in understanding the divergence of a function at a point (see
section 17.9), Stokes' theorem helps us understand what the Curl of a vector field is. Let P be a point on
the surface and Ce be a tiny circle around P on the surface. Then ∫ F • dr measures the amount of
Ce
circulation around P. You can see this by noticing that if F flows in the direction of the tangent vector,
then F • dr will be positive. If it flows in the opposite direction, then it will be negative. The stronger the
force field in the direction of the tangent vector, the greater the circulation, since the dot product of two
vectors has greatest magnitude when they point in the same direction. [More generally, ∫ F • dr , where C
C
is an oriented closed curve, is a measure of the tendency of a fluid (represented by F) to flow around C
and is called the circulation of F around C.]
Now, since the region enclosed by Ce is tiny, the surface integral can be approximated by
or (curl F) • N = Circulation per unit area = rate of circulation
So,
∫∫ (curl F) • NdS
represents the total measure of this circulation of a fluid (represented by F) over the
S
whole surface S and, according to Stoke, is equal to the tendency of a fluid to circulate around the rim of
the surface.
Example:
Consider the vector field F = yi + zj − xyk . Let S be the part of the plane z = 4 − x − 2 y in
the first octant and C be curve created when S intersects the coordinate planes. Use Stoke's
Theorem to find ∫ F • dr
C