Gauss divergence theorem for vectors can be easily explained by mass balance. But I can't think about one example for scalar gauss divergence theorem.
Gauss Divergence Theorem for scalars:
$$\int\limits_\Omega\nabla f(x)~\mathrm d\Omega=\int\limits_\Gamma f(x) \boldsymbol{n}~\mathrm d\Gamma,$$
Where:
$$f:\mathbb{\Omega}\to\mathbb{R},$$
$\Omega$ is an open set, such that $\Omega \subset \mathbb{R}^2$ (or $\mathbb{R}^3$)
$\Gamma$ is the boundary of $\Omega$ and $\boldsymbol n$ is the normal vector to $\Gamma$.
Answer
This theorem can be used to prove Archimede's Principle in a region with a non-uniform gravitational field.
The weight of the displaced fluid is $$\vec W=\int_\Omega \rho \vec g(\vec r)~\mathrm d\Omega.$$ Let us consider a body fully immersed. Then the buoyancy force is given by $$\vec B=-\oint_\Gamma p(\vec r)~\mathrm d\vec \Gamma =-\int_\Omega\vec\nabla p~\mathrm d\Omega,$$ where $p(\vec r)$ is the pressure, $d\vec \Gamma$ is the oriented area element and the given theorem is used.
A static fluid satisfy $$\vec\nabla p=\rho\vec g.$$ Plugging this into the integral for $\vec B$ and comparing with the integral for $\vec W$ we see that the buoyancy force equals the weight of the displaced fluid even when the gravitational field is non-uniform.
Edit
Proof of the result claimed by the OP:
Consider a vector field $f(x)\vec v$, where $f$ is a scalar field and $\vec v$ is a constant non vanishing vector. By the divergence theorem: $$\oint_\Gamma f\vec v\cdot ~\mathrm d\vec \Gamma=\int_\Omega \vec\nabla\cdot (f\vec v)~\mathrm d\Omega.$$ Since $\vec\nabla (f\vec v)=\vec v\cdot\vec\nabla f$, you get $$\vec v\cdot \oint_\Gamma f~\mathrm d\vec \Gamma=\vec v\cdot\int_\Omega\vec\nabla f~\mathrm d\Omega.$$ Since this holds for any constant vector $\vec v$ we get the result $$\oint_\Gamma f~\mathrm d\vec \Gamma=\int_\Omega\vec\nabla f~\mathrm d\Omega.$$ In fact I have seen people calling this the "gradient theorem".
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