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Math Help - divergence theorem problem

  1. #1
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    divergence theorem problem

    i hav another small question about calculus.please help...

    F= (x^3+2xy^2,y^3+2yz^2,z^3+2zx^2) is a vector field.

    fine integral( F.dS) on a sphere of radius R centered at the origin.

    I am not sure about what dS is. I think it should be a unit vector perpendicular to the surface u r calculating, is it right?
    I let n = ( 0, 0, 1) and try to integrate, but there are z and x , how to operate this?
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  2. #2
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    Let's say the sphere has radius a.

    (x^{3}+2xy^{2})i+(y^{3}+2yz^{2})j+(z^{3}+2zx^{2})k

    sphere x^{2}+y^{2}+z^{2}=a^{2}

    \frac{\partial}{{\partial}x}(x^{3}+2xy^{2})=x^{3}+  2xy^{2}

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

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

    Adding and factoring yields 5(x^{2}+y^{2}+z^{2})=5{\rho}^{2}

    5\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{a}{\rho}^{  4}sin{\phi}d{\rho}d{\phi}d{\theta}
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