I want to integrate $\int \nabla f(x)$ by substitution. Let $x = g(y)$.
Then is $\int \nabla f(x)$ equal to $\int \nabla [f(g(y))] |\det Dg| $ or $\int \nabla f|_{g(y)} |\det Dg| ?$
I am confused.
(PS: I know here I'm integrating a vector so think of it as $\nabla f \cdot \nabla f$ if you like.)