Use the divergence theorem to compute the net outward flux of $\vec F(x,y,z)=x^2\,\vec\imath-y^2\,\vec\jmath+z^2\,\vec k$ over the region $D$, where $D$ is the space between the planes $z=3-x-y$ and $z=9-x-y$ in the first octant.
Included is a plot of the region $D$.
By the divergence theorem, I know the flux is given by
$$2\iiint_D(x-y+z)\,\mathrm dx\,\mathrm dy\,\mathrm dz$$
and I can compute the integral easily enough (its value is $540$ by my calculation, confirmed both via divergence theorem and integration over each face), but I am wondering if there is a change of coordinates that I can employ to make the computation even easier. My instinct would be to transform $D$ into another triangular prism with uniform dimensions such that e.g. the triangular faces of the prism have the same area. Possibly something like
$$\begin{cases}x(u,v,w)=u\\y(u,v,w)=v\\z(u,v,w)=w-u-v\end{cases}$$
so that $3\le w\le9$ and the Jacobian $J$ is such that $|\det J|=1$, but I'm not sure what the other variables' bounds would be. Perhaps this change of variables is incorrect. Is there some way to make this work?