Let $G\subset \mathbb{R}^n$ be a an open bounded set with smooth boundary,$f:\text{closure(}G)\rightarrow \mathbb{R}$ which is $C^1$,and $u\in S^{n-1}$ constant vector.
Prove that $\int_{\partial G}<u,n>f(x)dS=\int_G\frac{\partial f}{\partial u} dx$ where $n(x)$ is then exterior unit normal to $\partial G$.
Can anyone please help?
You apply the divergence theorem link to the vectorial function $g(x)=f(x)u$.
Edit More details. You have $$\text{div} g=\sum_{i=1}^n\frac{\partial g_i}{\partial x_i}=\sum_{i=1}^n\frac{\partial f}{\partial x_i}u_i=\nabla f\cdot u=\frac{\partial f}{\partial u}$$ and $g\cdot n=f u\cdot n$.