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If $$F = \langle xy,yx,-(x+y)z \rangle$$ find $$\int F \cdot \hat n \, dS$$ where $S$ is a sphere of radius $2$ centered at $\langle 0,1,0 \rangle$?

I don't need a numerical solution to the problem because I know how to do it...for a sphere that is centered at the origin. How do I take into account the fact that the sphere is centered at $\langle 0,1,0 \rangle$? I am going to be using spherical coordinates, so can I simply add a '$-1$' onto the $y$ term of the parametrization? Or is this completely the wrong idea?

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Use the change of variables:

$$(x,y,x) \rightarrow (x,y-1,z)$$

Note also that $\nabla \cdot F = 0$ so by the Divergence Theorem the surface integral is $0$.

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