Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. F(x, y, z) = xyezi + xy2z3

Question

Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. F(x, y, z) = xyezi + xy2z3j − yezk, S is the surface of the box bounded by the coordinate plane and the planes x = 3, y = 6, and z = 1.

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Isabella 1 week 2021-09-09T11:51:28+00:00 1 Answer 0

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    2021-09-09T11:53:15+00:00

    \vec F has divergence

    \nabla\cdot\vec F=\dfrac{\partial(xye^z)}{\partial x}+\dfrac{\partial(xy^2z^3)}{\partial y}-\dfrac{\partial(ye^z)}{\partial z}=ye^z+2xyz^3-ye^z=2xyz^3

    By the divergence theorem, the integral of \vec F across S is equal to the integral of \nabla\cdot\vec F over the interior of S:

    \displaystyle\iiint_S\vec F\cdot\mathrm d\vec S=\int_0^1\int_0^6\int_0^32xyz^3\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac{81}2}

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