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pondzo
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Homework Statement
I am having trouble with part iii) of the following problem:
Verify the divergence theorem for the function ##\vec{u} = (xy,- y^2, + z)## and the surface enclosed by the three parts:
(i) ##z = 0, x^2+y^2 < 1##,
(ii) ##x^2+y^2 = 1, 0 \le z \le 1## and
(iii) ##(a^2-1)x^2+(a^2-1)y^2+z^2 = a^2 , 1 \le z \le a, a > 1##.
Homework Equations
Divergence theorem:
##\int_V\vec{ \nabla}\cdot\vec{u}~dV=\oint_{\partial v}\vec{u}\cdot\vec{dS}##
The Attempt at a Solution
The boundaries in iii) describe part of an ellipsoid. Rewrite the equation as follows:
##\frac{x^2}{(\frac{a}{\sqrt{a^2-1}})^2}+\frac{y^2}{(\frac{a}{\sqrt{a^2-1}})^2}+\frac{z^2}{a^2}=1##
LHS## = \int_0^{2\pi}\int_1^a\int_0^{\frac{z^2-a^2}{1-a^2}}\vec{ \nabla}\cdot\vec{u}~\rho~ d\rho ~dz ~d\theta##
##~~~~ = \int_0^{2\pi}\int_1^a\int_0^{\frac{z^2-a^2}{1-a^2}}\rho - \rho^2\sin{\theta}~ d\rho ~dz ~d\theta##
##~~~~=\frac{4\pi}{1-a^2}##
I think that is correct... maybe someone can check my setup.
I'm stuck with the RHS.. I tried to use cylindrical coordinates and it got ugly quickly, I tried to use generalised spherical coordinates but it got ugly equally as fast. I expect the process to be somewhat ugly, but mine is so much I feel for certain I'm doing something wrong. I'll give a brief overview of my strategy:
For cylindrical coordinates I tried to parameterize it as follows: ##\vec{r}(\theta,z)=(\frac{z^2-a^2}{1-a^2}\cos{\theta},\frac{z^2-a^2}{1-a^2}\sin{\theta},z)##. For starters I am not even sure if this is right. I then get the normal vector to this by computing: ##\vec{n}=\frac{\partial \vec{r}}{\partial \theta}\times\frac{\partial \vec{r}}{\partial z}##. Computing this and normalising i get: ##\hat{n}=\frac{(\cos{\theta},\sin{\theta},\frac{-2z}{1-a^2})}{\sqrt{1+\frac{4z^2}{(1-a^2)}}}##.Plugging all of this into the flux integral looks extremely messy and I don't think I will be able to do the integral. So I abort and resort to spherical coordinates...
For spherical coordinates I tried to parameterize it as follows: ##\vec{r}(\theta,\phi)= (\frac{a}{\sqrt{a^2-1}}\sin{\theta}\cos{\phi},\frac{a}{\sqrt{a^2-1}}\sin{\theta}\sin{\phi},a\cos{\theta})## and the rest of the story goes like the cylindrical one..Any help would be greatly appreciated!
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