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

Accepted Solution

[tex]\vec F[/tex] has divergence[tex]\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[/tex]By the divergence theorem, the integral of [tex]\vec F[/tex] across [tex]S[/tex] is equal to the integral of [tex]\nabla\cdot\vec F[/tex] over the interior of [tex]S[/tex]:[tex]\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}[/tex]