Integrate a partial derivative.

Question

I have seen that $\int_{[a,b]}f'(x)=f(b)-f(a)$ by the fundamental theorem of calculus. But what is $\int_{\Omega}\frac{\partial}{\partial x_i}f(x_1,\ldots,x_n)$?

Answer 1

If $\Omega$ is smooth enough, then this is

$$ \int_{\partial\Omega} f(\sigma) \nu_i(\sigma) \,d \sigma $$

where $\nu=(\nu_1,\ldots,\nu_N)$ is the outward pointing unit normal to $\partial\Omega$.