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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
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This note was uploaded on 04/18/2008 for the course MATH 234 taught by Professor Boeri during the Winter '08 term at Northwestern.
 Winter '08
 Boeri
 Multivariable Calculus

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