I came across the following theorem:
$$ \iint_{\partial \Omega}f dS=\iiint_{\Omega}\nabla{f}dV $$ $\Omega$ is a bounded region whose boundary $\partial \Omega$ is a closed, piecewise smooth surface which is oriented with an outward normal direction and $f$ is a $C^1$ function in an open region that contains $\Omega$
But I guess I quite don't understand the concept of volume integral of a vector field and it could be that a scalar obtained in the LHS equals a vector obtained in the RHS. Thanks in advance.