Dina and Masha started out on a 20 miles bike path at the same time. When Dina reached the end of the 20 miles, Masha still had
2 answers:
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Looks like we have
which has divergence
By the divergence theorem, the integral of across
is equal to the integral of
over
, where
is the region enclosed by
. Of course,
is not a closed surface, but we can make it so by closing off the hemisphere
by attaching it to the disk
(call it
) so that
has boundary
.
Then by the divergence theorem,
Compute the integral in spherical coordinates, setting
so that the integral is
The integral of across
is equal to the integral of
across
plus the integral across
(without outward orientation, so that
Parameterize by
with and
. Take the normal vector to
to be
Then we have
Finally,