Lesson 25: Stokes’ Theorem
July 31th, 2015
Section 16.8
Green’s Theorem tells us that the double integral of a vector
field over a surface imbedded in the plane is equal to the
integral of that vector field over the bounding plane curve.
Today we will look at Stokes’ Theorem. Stokes’ Theorem
relates the integral of a vector field over a general surface in
3-dimensional space with the integral of that vector field over
the bounding space curve.
More succinctly, both theorems compare two-dimensional
regions to one-dimensional regions, but Green’s Theorem deals
with surfaces in the plane and plane curves while Stokes’
Theorem deals with surfaces in space and space curves.
Section 16.8
When applying Stokes’ Theorem, we will orient S with its unit
normal vector n. We say that this orientation induces the
positive orientation of the boundary curve C .
To visualize this, if you walk in the positive direction around C
with your head pointing in the direction of n, then the surface
will always be on your left.
Section 16.8
Theorem (Stokes’ Theorem)
Let S be an oriented piecewise-smooth surface 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 region in R3
that contains S. Then
Z
C
F · dr =
ZZ
curl F · dS
S
Section 16.8
Some comments:
Since
Z
C
F·dr =
Z
C
F·T ds
and
ZZ
curl F·dS =
S
ZZ
curl F·n dS
S
Stokes’ Theorem tells us that the integral of the
tangential component of F over the boundary curve of S
is equal to the integral of the normal component of curl F
over S itself.
Section 16.8
The positively oriented boundary curve of S is often
written ∂S. Then Stokes’ Theorem is written as
ZZ
curl F · dS =
S
Z
F · dr
∂S
and we again see an analogy between Stokes’ Theorem,
Green’s Theorem, and the Fundamental Theorem of Line
Integrals. The left hand side has the integral of (in a
sense) the derivative of our function over a region, and
the right hand side deals with values of that function on
the boundary of that region.
Section 16.8
Finally, if S is imbedded in the xy -plane, then the unit
normal vector for S is just k, and so Stokes’ Theorem
becomes
Z
F · dr =
C
ZZ
curl F · dS =
ZZ
S
(curl F) · k dA
S
This last integral is just the vector form of Green’s
Theorem, i.e.
ZZ
(curl F) · k dA =
S
ZZ
(
S
∂Q ∂P
−
) dA
∂x
∂y
This shows that Green’s Theorem is in fact a special case
of Stokes’ Theorem.
Section 16.8
Example
Evaluate C F · dr, where F(x , y , z) = −y 2 i + x j + z 2 k and C
is the curve of intersection of the plane y + z = 2 and the
cylinder x 2 + y 2 = 1. (Orient C to be counterclockwise when
viewed from above.)
R
Section 16.8
In the previous example, we used Stokes’ Theorem to compute
a line integral by turning it into a simple surface integral. As
with all of these things, we can and will want to go in the
reverse direction as well.
Section 16.8
Example
Use Stokes’ Theorem to compute the integral S curl F · dS,
where F(x , y , z) = xz i + yz j + xy k and S is the part of the
sphere x 2 + y 2 + z 2 = 4 that lies inside the cylinder
x 2 + y 2 = 1 and above the xy -plane.
RR
Section 16.8
In the previous example, we computed a surface integral over a
surface S in a way that only required us to know the values of
our vector field F on the boundary curve C of that surface. It
follows that if we took another surface S1 with the same
boundary curve (satisfying the conditions of Stokes’ Theorem),
we would get the same answer. Thus for two surfaces S1 and
S2 with the same boundary curve C , we have
ZZ
curl F · dS =
S1
Z
F · dr =
ZZ
C
curl F · dS
S2
This means that if we have to compute a surface integral over
a complicated surface, sometimes we can get away with
computing an integral over a much simpler surface.
Section 16.8
Example
Evaluate S curl F · dS, where F(x , y , z) = e xy i + e xz j + x 2 z k
and S is the half of the ellipsoid 4x 2 + y 2 + 4z 2 = 4 that lies
to the right of the xz-plane, oriented in the direction of the
positive y -axis.
RR
Section 16.8
Stokes’ Theorem can help us understand the curl vector.
Imagine fluid flowing around an oriented closed curve C with
velocity vector v. We have the line integral
R
C
v · dr =
R
C
v · T ds
Since v · T is the tangential component, it is larger theR closer
that v and T are to being in the same direction. Thus C v · dr
measures the tendency of fluid to move around C and is called
the circulation of v around C .
Section 16.8
Now consider a point P0 (x0 , y0 , z0 ) in the fluid and let Sa be a
small disc of radius a centered at P0 . Since curl F is
continuous, we have (curl F)(P) ≈ (curl F)(P0 ) for all points P
in Sa . Letting Ca be the boundary circle of Sa , Stokes’
Theorem gives
Z
v · dr =
Ca
ZZ
curl v · dS =
Sa
≈
ZZ
Sa
ZZ
curl v · n dS
Sa
curl v(P0 ) · n(P0 ) dS = curl v(P0 ) · n(P0 )πa2
As a → 0 we have
1 Z
v · dr
a→0 πa 2 Ca
curl v(P0 ) · n(P0 ) = lim
So curl v · n measures the rotating effect of the fluid around n,
as we claimed in Lesson 21.
Section 16.8
Example
Evaluate C F · dr, where F(x , y , z) = xy i + 2z j + 3y k and C
is the curve of intersection of the plane x + z = 5 and the
cylinder x 2 + y 2 = 9.
R
Section 16.8