Normally I read the Divergence Theorem written as (\oiint doesn't exist here):
\begin{align}
\oint_{\partial \Omega} \vec{F} \cdot \hat{n} \; dS = \iiint_{\Omega} \nabla \cdot \vec{F} \; dV
\end{align}
Where $\partial \Omega$ is the boundary of region $\Omega$ and $dS$ are infinitesimal surface of $\partial \Omega$ and $dV$ infinitesimal volume of $\Omega$.
But reading a paper CASTILLO and GRONE, A matrix analysis approach to higher order approximations for divergence and gradients satisfying global conservation law, it states the Divergence Theorem as:
\begin{align}
\int_{\partial \Omega} f \, \vec{v} \cdot \hat{n} \; dS = \int_{\Omega} \nabla \cdot \vec{v} \, f \; dV + \int_{\Omega} \vec{v} \, \nabla f \; dV
\end{align}
Are those equivalent? if so how can I correlate the RHS of both equations?
I really think that he is abusing notation using a single integral instead a triple one and not directly specifying the loop integral in left hand side, but even with that I cannot correlate, maybe I'm missing some definition here.
Best Answer
The only error here is the second integral on the RHS should have the dot product as the integrand:
$$ \int_{\Omega} \vec{v} \cdot \, \nabla f \; dV, $$
since $\nabla \cdot (f \,\vec{v}) = f \, \nabla \cdot \vec{v} + \nabla f \cdot \vec{v}$.
Aside from appearance, given an integrable function $f:\Omega \subset \mathbb{R}^3 \to \mathbb{R}$ there is no difference between
$$\int_\Omega f, \quad \int_\Omega f \, dV, \quad\text{and } \quad \int\int\int_\Omega f \, dV$$
There is difference between a multiple Riemann integral and an iterated Riemann integral, for example,
$$\int_{[0,1]^3} f \quad \text{and} \quad\int_0^1 \left(\int_0^1 \left(\int_0^1 f(x,y,z) \, dz \right)\, dy \right) \, dx$$
where the object on the right may exist but the object on the left may not. However, if $|f|$ is integrable on $[0,1]^3$, then the two must be equal, as one consequence of Fubini's theorem.