Question

Use the divergence theorem to compute the net outward flux of the field f=2x, y, 3z across the surface s, where s is the sphere (x,y,z): x2 y2 z2=6.

Answer 1

Given the field

`\vec F(x,y,z) = \langle2x,y,3z\rangle`

Compute its divergence.

`\mathrm{div}\vec F(x,y,z)= (\partial(2x))/(\partial x) + (\partial y)/(\partial y) + (\partial(3z))/(\partial z) = 2 + 1 + 3 = 6`

By the divergence theorem, the flux of
`\vec F` across
`\vec S` is

`\displaystyle \iint_S \vec F \cdot d\vec S = \iiint_{\mathrm{int}(S)} \mathrm{div}\vec F(x,y,z)\,dV = 6 \iiint_{\mathrm{int}(S)} dV`

which is simply 6 times the volume of
`S`. (By
`\mathrm{int}(S)`, I mean the "interior of
`S`".)

`S` has radius √6, so its volume is 4/3 π (√6)³ = 8√6 π, and the flux is

`\displaystyle \iint_S \vec F\cdot d\vec S = 6 \cdot 8\sqrt6\,\pi = \boxed{48\sqrt6\,\pi}`