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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 $\Gamma$.

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closed as unclear what you're asking by ACuriousMind, Gert, CuriousOne, Cosmas Zachos, knzhou Jul 14 '16 at 22:47

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ This question (v8) seems like a list question. $\endgroup$ – Qmechanic Jul 14 '16 at 2:58
  • $\begingroup$ @Qmechanic Yes, it can be interpreted as a list question, but can you understand its importance? If yes, how could it be stated without being a list question like? I was looking for just one good example and I got one. $\endgroup$ – Vitor Abella Jul 14 '16 at 3:06
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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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