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rasbuurry
29.10.2019 •
Mathematics
Problem 4: let f = (2z + 2)k be the flow field. answer the following to verify the divergence theorem: a) use definition to find the upward flux (normal vector pointing upward in z direction) along the hemisphere given in problem 1(b). b) use definition to find the downward flux on the circle x 2 + y 2 ≤ 3 on the xy-plane. c) using the results from parts (a) and (b), what is the net flux over the closed northern hemisphere? d) use the divergence theorem to verify that this answer agrees. [hint: dv = rho 2 sin ϕ drho dθ dϕ.]
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Ответ:
Given that you mention the divergence theorem, and that part (b) is asking you to find the downward flux through the disk
, I think it's same to assume that the hemisphere referred to in part (a) is the upper half of the sphere
.
a. Let
denote the hemispherical cap
, parameterized by
with
and
. Take the normal vector to
to be
Then the upward flux of
through
is
b. Let
be the disk that closes off the hemisphere
, parameterized by
with
and
. Take the normal to
to be
Then the downward flux of
through
is
c. The net flux is then
.
d. By the divergence theorem, the flux of
across the closed hemisphere
with boundary
is equal to the integral of
over its interior:
We have
so the volume integral is
which is 2 times the volume of the hemisphere
, so that the net flux is
. Just to confirm, we could compute the integral in spherical coordinates:
Ответ:
I guess you forget to add the picture