Question

(a) find the flux of f out of the rectangular solid 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤

c. your answer will be in terms of a, b,
c. flux =

Answer 1

Without knowing the details of the vector field, I can't give you a direct answer for the flux. At any rate, the general idea would be to use the divergence theorem, which states the flux of
`\mathbf f` over the closed surface
`\mathcal S` (surface of a cube in this case) is equivalent to the integral of the divergence of
`\mathbf f` over the interior of the surface (call it
`\mathcal R`):


`\displaystyle\iint_(\mathcal S)\mathbf f\cdot\mathrm d\mathbf S=\iiint_(\mathcal R)(\\abla\cdot\mathbf f)\,\mathrm dV`

The latter integral is less work to compute, and hence the usefulness of the divergence theorem. Denoting the vector field by
`\mathbf f(x,y,z)=(f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))`, we have


`\displaystyle\iiint_(\mathcal R)(\\abla\cdot\mathbf f)\,\mathrm dV=\int_(z=0)^(z=c)\int_(y=0)^(y=b)\int_(x=0)^(x=a)\left((\partial f_1)/(\partial x)+(\partial f_2)/(\partial y)+(\partial f_3)/(\partial z)\right)\,\mathrm dx\,\mathrm dy\,\mathrm dz`