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bitrex
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Homework Statement
I have to evaluate the surface integral of the following function over the top hemisphere of a sphere.
Homework Equations
[tex]\sigma (x,y,z) = \frac{\sigma_0 (x^2+y^2)}{r^2}[/tex]
[tex]z = \sqrt{r^2-x^2-y^2}[/tex]
[tex]\iint G[x,y, f(x,y)] \sqrt{1+ \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}}dx dy[/tex]
The Attempt at a Solution
[tex]\frac{\partial f}{\partial x} = \frac{-x}{z}[/tex]
[tex]\frac{\partial f}{\partial y} = \frac{-y}{z}[/tex]
so
[tex] \sqrt{1 + \frac{x^2}{z^2} + \frac{y^2}{z^2}} = \frac{r}{z} [/tex]
since [tex]x^2+y^2+z^2 = r^2[/tex]
So multiplying I get
[tex] \frac{\sigma_0(x^2+y^2)r}{r^2(r^2-x^2-y^2)^{\frac{1}{2}}}[/tex]
The problem is when I try to convert this into polar coordinates to do the double integral by substituting [tex]x = r cos \theta, y = r sin \theta[/tex] the stuff in the lower parenthesis becomes zero which is obviously a problem. Where am I going wrong here? Thanks for any help.
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