What's the difference between Gauss' Theorem and Stokes' Theorem? Does Gauss's Theorem take an integral over an "inner product" derivative while Stokes's Theorem takes an integral over an exterior derivative? And is "divergence" associated with Gauss's Theorem and "curl" associated with Stokes's Theorem? And does "divergence" refer to movements of (e.g. fluids) TO (and from) a surface, while curl refers to movements AROUND a surface)?
[Math] Gauss’s Theorem vs. Stokes’s Theorem
multivariable-calculus
Best Answer
Before giving a comparison/contrast type answer, let's first examine what the two theorems say intuitively.
Stokes' Theorem says that if $\mathbf{F}(x,y,z)$ is a vector field on a 2-dimensional surface $S$ (which lies in 3-dimensional space), then $$\iint_S \text{curl }\mathbf{F}\cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F}\cdot d\mathbf{r},$$ where $\partial S$ is the boundary curve of the surface $S$.
The left-hand side of the equation can be interpreted as the total amount of (infinitesimal) rotation that $\mathbf{F}$ impacts upon the surface $S$. The right-hand side of the equation can be interpreted as the total amount of "spinning" that $\mathbf{F}$ affects along the boundary curve $\partial S$. Stokes' Theorem then tells us that these two seemingly different measures of "spin" are in fact the same!
It is remarkable also because solely from knowing how $\mathbf{F}$ affects the boundary curve $\partial{S}$, we can deduce how $\text{curl }\mathbf{F}$ affects the entire surface!
The Divergence Theorem says that if $\mathbf{F}(x,y,z)$ is a vector field on a 3-dimensional solid region $E$ (which lies in 3-dimensional space), then $$\iiint_E \text{div }\mathbf{F}\,dV = \iint_{\partial E} \mathbf{F}\cdot\mathbf{N}\,dS,$$ where $\partial E$ is the boundary surface of the solid region $E$, and $\mathbf{N}$ is an outward-pointing normal vector field on $E$.
If we think of $\mathbf{F}$ as being some sort of fluid, then the left-hand side measures how much of the fluid is outward-flowing (like a source) or inward-flowing (like a sink). That is, the left-hand side measures the total amount of (infinitesimal) divergence (outwardness/inwardness) of $\mathbf{F}$ throughout the entire solid $E$.
On the other hand, the right-hand side tells us how much of $\mathbf{F}$ is "passing through" the boundary surface $\partial E$. In other words, it is the flux of $\mathbf{F}$ across $\partial E$.
So, the Divergence Theorem tells us that these two different measures of the "outwardness" of $\mathbf{F}$ (the sources/sinks inside the solid vs the flux through the boundary) are in fact the same! To quote Wikipedia: "The sum of all sources minus the sum of all sinks gives the net flow out of a region."
And again, we have a situation where the behavior of $\mathbf{F}$ on the boundary gives us insight into how $\mathbf{F}$ acts on the entire region!
Similarities: Both Stokes' Theorem and the Divergence Theorem relate behavior of a vector field on a region to its behavior on the boundary of the region. As Zhen Lin pointed out in the comments, this similarity is due to the fact that both Stokes' Theorem and the Divergence Theorem are but special cases of a single, very powerful equation (known as the Generalized Stokes Theorem).
(The Generalized Stokes Theorem is somewhat advanced, and usually goes by the name Stokes' Theorem, whereas the Stokes' Theorem we've been talking about is often called the Kelvin-Stokes Theorem. This is why the Wikipedia page on "Stokes' Theorem" may seem rather advanced -- it is primarily about the Generalized theorem.)
Differences: Stokes' Theorem talks about "rotation" along a surface which has a boundary curve. The Divergence Theorem talks about "sources and sinks" inside a solid that has a boundary surface.
So, in addition to being about different types of quantities ("rotation" vs "divergence"), you should note that the two theorems apply to completely different types of regions. That is, a surface which has a boundary curve (setting of Stokes' Theorem) cannot enclose a solid volume (setting of the Divergence Theorem), and conversely.