Gauss's Theorem vs. Stokes's 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)?

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I'm not sure what exactly you're asking... In Gauss' Theorem there is an expression involving the divergence, and in Stokes' Theorem there is an expression involving the curl, yes. Do you mean to ask whether there is any connection between the two theorems? Or do you want an intuitive explanation of why the theorems should be true? –  Jesse Madnick Jun 27 '11 at 3:38
@Jesse: I wanted a connection/contrast type answer. That is, does one have an "inner product" outcome, and the other, an "exterior product" outcome? –  Tom Au Jun 27 '11 at 13:42

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.

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By the way, the Fundamental Theorem of Calculus $$\int_a^b f'(x) = f(b) - f(a)$$ is also a special case of the Generalized Stokes Theorem, and relates the integral over an interval $[a,b]$ to the values of the function on the boundary -- the boundary being the two points $x = a$ and $x = b$. –  Jesse Madnick Jun 28 '11 at 3:08
in the mentioned theorems, can the $\partial E$ be non differentiable everywhere or should the boundary surface be sufficiently smooth? –  pluton Jan 20 '13 at 0:18

For a typical vector calculus course, curl is associated with Stokes' Theorem, while the divergences is associated with Gauss' theorem.

I believe the first paragraph in each of the following two links will answer some other questions you might have:

Gauss Theorem.

Stokes' Theorem.

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I DID consult these two links, and came away more confused than before. –  Tom Au Jun 27 '11 at 0:39
@Tom Au: What exactly do you find confusing? Please be specific. –  Jesse Madnick Jun 27 '11 at 5:19
@jesse madnick: I am a "liberal arts" type who is big on analogies. So I get something like, Gauss' Theorem uses an inner product, and therefore "divergence" is moving to and from a surface, while Stokes' theorem uses a a cross product, so "curl," or "rotation"is moving AROUND the surface." –  Tom Au Jun 27 '11 at 13:48
@Tom Au: Actually, there's a much better way to look at it: these two theorems are actually special cases of the generalised Stokes theorem. So the analogy runs a lot deeper than that. –  Zhen Lin Jun 27 '11 at 14:13

In one sense, if you associate curl with Stokes and divergence with the Divergence theorem, you have what you need to recognize when to use each. But instead of a mnemonic, I like to hold onto an example, to see the different applications.

Gauss's theorem is frequently employed in electrodynamics. With it, you get the fact that the electric flux out of a volume of space is related to the distribution of the field within that space (as measured by divergence), which is in turn related to the amount of charge in that space. See here.

Stokes is also used is electrodynamics, but it is instead more directly used on Ampere's Law. It relates magnetic fields to the current that creates them, and this is sort of nice in the sense that it gives a reason why we should traverse a shape in a loop.

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