Prove $\unicode{x222F}_S{f(x,y,z)d\vec{S}}=\iiint_E{}\nabla f(x,y,z)dV$.

Question

Let the closed surface $S$ be delimit volume $E$ and let the scalar function $f$. I need to demonstrate that :

$$\unicode{x222F}_S{f(x,y,z)d\vec{S}}=\iiint_E{}\nabla f(x,y,z)dV$$

I do have some indication. First, I got to consider the vector field $\vec{F}=f(x,y,z)\vec{u}$ where $\vec{u}$ has components $u_1, u_2, u_3$ independent of $x,y,z$ and I need to use divergence theorem on $S$. Next, I got to use the theorem that if $\vec{a}\cdot\vec{c}=\vec{b}\cdot\vec{c}$ then $\vec{a}=\vec{b}$.

Unfortunately, these hints doesn't really help and I can't figure out how to demonstrate this result. The demonstration is supposed to take 1 or 2 lines only !

Thanks !

Answer 1

This is the proof Divergence Theorem.

bests!