Lizzie Wagner
Stoke’s Theorem
Stoke’s theorem is a
three
dimensional
elaboration of Green’s theorem.
In
Green’s theorem, we related a line integral to a double integral over some region
R
. In
Stoke’s theorem, we are going to relate a line integral to a surface integral. However,
please note that in this theorem the surface
S
can actually be any surface as long as its
boundary curve is given by
C
. Stoke’s theorem states that the closed line integral over
C
of a vector field
F
dotted with the positive unit tangent vector
T
to the boundary curve
C
is equal to the surface integral (calculated with the use of double integration) of the
normal component of the curl of that vector field ((gradient cross
F
) dotted with
n
) over
some surface
S
.
The full theorem is shown below.
∫
C
F
.
dr
=
∫
C
F
.
T
d
s
=
∫ ∫
S
(curl
F)
.
n
d
S
For me, this is the trickiest theorem to get a good grasp on.
To use this theorem,
you must remember to parameterize the surface prior to computing anything.
Also, you
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Boeri
 Multivariable Calculus, Vector Calculus, Stokes' theorem, Unit Normal Vector, C. Stoke

Click to edit the document details