Calculating flux from an inwards orientation

Question

I have to calculate the flux between two paraboloids (so an enclosed region), but oriented inwards. I calculated the normal vectors, but found that the integration required was super hard. I realized that using the divergence theorem would make things a lot easier, but I am confused as to how the divergence theorem accounts for an inward or outward orientation.

I have always used the divergence theorem to calculate flux of a region with an outwards orientation. Is it as simply as multiplying by -1 to the flux or am I missing something.

Answer 1

Flux integrals are defined by calculating how much of the vector $\vec{F}(x,y,z)$ is in the direction of $n$. The divergence theorem, on the other hand, is proven using flux integrals, which are assumed to be oriented outwards.

When trying to calculate inward flux, we can simply choose a normal vector that is oriented inward relative to the surface if we were to use a flux integral. If one wanted to calculate this using divergence theorem, then it simply requires multiplying the answer by $-1$ after completing the integral.