Question

Compute the flux of the vector field LaTeX: \vec{F}=F → =< y + z , x + z , x + y > though the unit cubed centered at origin.

Answer 1

Assuming the cube is closed, you can use the divergence theorem:

`\displaystyle\iint_S\vec F\cdot\mathrm dS=\iiint_T\mathrm{div}\vec F\,\mathrm dV`

where
`S` is the surface of the cube and
`T` is the region bounded by
`S`.

We have

`\mathrm{div}\vec F=(\partial(y+z))/(\partial x)+(\partial(x+z))/(\partial y)+(\partial(x+y))/(\partial z)=0`

so the flux is 0.