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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 = 7, y = 2, and z = 1.

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LammettHash

By the divergence theorem, the surface integral of [tex]\vec F[/tex] across the closed surface [tex]S[/tex] is equal to the integral of the divergence of [tex]\vec F[/tex] over the region [tex]R[/tex] enclosed by [tex]S[/tex].

The divergence of [tex]\vec F[/tex] is

[tex]\nabla\cdot\vec F=ye^z+2xyz^3-ye^z=2xyz^3[/tex]

Then the flux of [tex]\vec F[/tex] across [tex]S[/tex] is

[tex]\displaystyle\iiint_R(\nabla\cdot\vec F)\,\mathrm dV=\int_0^1\int_0^2\int_0^72xyz^3\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{\frac{49}2}[/tex]

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