Question

divergence theorem help please.

Answer 1

F(x,y,z)=<x^3,y^3,z^3> ....s is surface of the region that is emclosed by hemisphere z= (a^2-x^2-y^2)^(1/2) and z=0

Answer 2

I will be back for this problem in about 20 min

Answer 3

\[\nabla\cdot \vec F(x,y,z)=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}\]

Answer 4

*

Answer 5

\[\vec F(x,y,z)=\langle x^3,y^3,z^3\rangle\]\[\iint\limits_D\vec F\cdot d\vec S=\iiint\limits_E\text{div}\vec FdV\]\[\text{div}\vec F=\nabla\cdot\vec F=3x^2+3y^2+3z^2\]\[S=z= (a^2-x^2-y^2)^{1/2}\text{ and }z=0\]\(E\) should be in polar coordinates because of symmetry, so we get (from the fact that this is the upper half of a sphere of radius \(a\) )\[0\le\rho\le a\]\[0\le\phi\le \frac\pi2\]\[0\le\theta\le2\pi\]\[dV=\rho^2\sin\phi d\rho d\phi d\theta\]transformation to spherical coordinates is\[x=\rho\sin\phi\cos\theta\]\[y=\rho\sin\phi\sin\theta\]\[z=\cos\phi\]so we get\[\iint\limits_D\vec F\cdot d\vec S=\iiint\limits_E\text{div}\vec FdV\]\[=\int_0^{2\pi}\int_0^{\frac\pi2}\int_0^a3\rho^2(\sin^2\phi\cos^2\theta+\sin^2\phi\sin^2\theta+\cos^2\phi)\rho\sin\phi d\rho d\phi d\theta\]\[=\int_0^{2\pi}\int_0^{\frac\pi2}\int_0^a3\rho^3(\sin^2\phi+\cos^2\phi)\sin\phi d\rho d\phi d\theta\]\[=\int_0^{2\pi}\int_0^{\frac\pi2}\int_0^a3\rho^3\sin\phi d\rho d\phi d\theta\]I hope you can do that integral; it shouldn't be the hard part of the problem for you
enjoy :)

Answer 6

PS: the only reason I give the full solution is because
1) you are gone
2)this is a tricky problem for many
normally I would only help you get close, but her I trust you to figure out what I did

Answer 7

I get\[\frac{3\pi a^4}2\]for the final answer.

Answer 8

typo, I meant that
*\(E\) should be in spherical coordinates*
oh well, the work is still right