Two people enter a whispering gallery and want to know where they should stand so they can hear the other person's whisper. The
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Treat 
as the boundary of the region 
, where is the part of the surface 
bounded by . We write
with
.
By Stoke's theorem, the line integral is equivalent to the surface integral over of the curl of 
. We have
so the line integral is equivalent to
where
is a vector-valued function that parameterizes . In this case, we can take
with
and 
. Then
and the integral becomes
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with
By Stoke's theorem, the line integral is equivalent to the surface integral over
so the line integral is equivalent to
where
with
and the integral becomes