78
C
HAPTER
5
T
HREE
D
IMENSIONAL
A
NALOGUES
C
HANGING THE SURFACE
Stoke’s Theorem tells us that the circulation of
F
across a surface
S
is determined solely
by the values of
F
along the boundary of the surface,
S
. Therefore the circulation of
F
across
S
is the same as the circulation of
F
across
any
surface with the same boundary,
so if neither the surface integral nor the line integral form of Stoke’s Theorem simplify
into manageable integrals, we have one more trick at our disposal: we can compute the
surface integral across a nicer surface with the same boundary. The following example
demonstrates this approach.
Example 1.
Compute the circulation of
F
ye
xy
y
3
3
, x
2
y
xe
xy
,
sin
xy
across the upper hemisphere of the
3
d unit sphere, that is, the surface
S
x, y, z
:
x
2
y
2
z
2
1
and
z
0
.
Solution.
First notice that computing circulation as a line integral is going to be impos
sible. The boundary of
S
is the unit circle, so parameterizing this in the standard way, we
have
r
t
cos
t,
sin
t,
0
,
r
t
sin
t,
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 Fall '08
 Keeran
 Calculus, Cos, Line integral, dr dθ, Stokes' theorem

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