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Math Help - Divergence theorem question

  1. #1
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    Divergence theorem question

    Calculate the flux of F across S.

    Let F = \frac{r}{|r|} where r=x\vec{i}+y\vec{j}+z\vec{k}, S consists of the hemisphere z=\sqrt{1-x^2-y^2} and the disk x^2+y^2 \leq 1 in the xy plane

    so for F I got:

    \frac{x\vec{i}+y\vec{j}+z\vec{k}}{\sqrt{x^2+y^2+z^  2}}

    therefore my divergence will be:

    \frac{\partial}{\partial x} = \frac{y^2+z^2}{(x^2+y^2+z^2)^{3/2}}

    \frac{\partial}{\partial y} = \frac{x^2+z^2}{(x^2+y^2+z^2)^{3/2}}

    \frac{\partial}{\partial z} = \frac{x^2+y^2}{(x^2+y^2+z^2)^{3/2}}

    now putting everything together I get:

    \frac{2(x^2+y^2+z^2)}{(x^2+y^2+z^2)^{3/2}}= \frac{2}{\sqrt{x^2+y^2+z^2}}

    now converting into cylindrical coordinates I get:

    \int_{\theta=0}^{\theta=2\pi} \int_{\phi=0}^{\phi=\pi/2} \int_{\rho=0}^{\rho=1} = \frac{2\rho^2 \sin(\phi)}{\rho} \ d\rho \ d\phi \ d\theta = 2\pi

    now I don't know if this is correct since x^2+y^2 \leq 1 is throwing me off.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Calculate the flux of F across S.

    Let F = \frac{r}{|r|} where r=x\vec{i}+y\vec{j}+z\vec{k}, S consists of the hemisphere z=\sqrt{1-x^2-y^2} and the disk x^2+y^2 \leq 1 in the xy plane

    so for F I got:

    \frac{x\vec{i}+y\vec{j}+z\vec{k}}{\sqrt{x^2+y^2+z^  2}}

    therefore my divergence will be:

    \frac{\partial}{\partial x} = \frac{y^2+z^2}{(x^2+y^2+z^2)^{3/2}}

    \frac{\partial}{\partial y} = \frac{x^2+z^2}{(x^2+y^2+z^2)^{3/2}}

    \frac{\partial}{\partial z} = \frac{x^2+y^2}{(x^2+y^2+z^2)^{3/2}}

    now putting everything together I get:

    \frac{2(x^2+y^2+z^2)}{(x^2+y^2+z^2)^{3/2}}= \frac{2}{\sqrt{x^2+y^2+z^2}}

    now converting into cylindrical coordinates I get:

    \int_{\theta=0}^{\theta=2\pi} \int_{\phi=0}^{\phi=\pi/2} \int_{\rho=0}^{\rho=1} = \frac{2\rho^2 \sin(\phi)}{\rho} \ d\rho \ d\phi \ d\theta = 2\pi

    now I don't know if this is correct since x^2+y^2 \leq 1 is throwing me off.
    Looks fine to me. The disc (the bottom of the hemisphere) is described by x^2+y^2 \le 1 but since your using the divergence thm, you really don't consider this part of the surface but instead the volume as a whole.
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