TITLE: When integrating how do I choose wisely between Green's, Stokes' and Divergence?
QUESTION [24 upvotes]: I've been taught Green's Theorem, Stokes' Theorem and the Divergence Theorem, but I don't understand them very well. In particular I don't understand in what circumstances I would choose to use any one of these over any one of the others. What criteria should I be looking for to help me decide? I specifically want to use the theorems to make integration as easy or trivial as possible.

REPLY [17 votes]: All these theorems have the same property: namely it relates the integral over some type of space to the integral over the boundary of that space. In the following examples $A$ is some kind of space (2, 3, 4, etc dimensional) and $\partial A$ represents its boundary. For example, if $A$ is a surface, then its boundary is a line. If $A$ is a volume, its boundary is a surface. If $A$ is 4-dimensional, then its boundary could be a 3-dimensional volume.
Here are some particular examples:

Green's Theorem  (we turn the double integral over a SURFACE to a line integral around its BOUNDARY, a line):
$$\int\int_A \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dxdy = \oint_{\partial A} (Ldx + Mdy)$$ 
Divergence Theorem (we turn the triple integral over a VOLUME into a surface integral around its BOUNDARY, a surface):

$$\int\int\int_V (\nabla \cdot F) dV = \int\int_{\partial V} (F \cdot n) dS$$
An instructive example follows:

the Fundamental theorem of calculus (here we turn an integral over an INTERVAL into an "integral" over its BOUNDARY, the endpoints (integration on the right is called "integration over $0$-forms", it's written this way to emphasize the use of the boundary of $[a,b]$ in the regular fundamental theorem of calculus):
$$\int_a^b f'(x) dx = f(b) - f(a) = \int_{\{b,-a\}} f'$$

All of these theorems have the same "general form" called "the general Stokes' theorem":
$$\int_A df = \int_{\partial A} f$$
where $df$ is defined appropriately.
All of the integrands in these theorems are related to each other by what's called the "exterior derivative" of the "differential form". The "differential form" is literally your integrand (including the dx) and the calculus of differential forms explains how you perform a computation to get from one type of integrand to another (how you get from $\left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dxdy$ to $Ldx + Mdy$ in the case of Green's Theorem).
The details here aren't important, but it should illustrate to you that to use these theorems, you need to 

Figure out which one you are "in" by writing it in a form that matches those in the theorems.
Figure out which one is related to the one you are "in", i.e. the other side of the "=" sign to the one you are "in".
Rewrite what you have with respect to the one you are related to, usually by taking a derivative or finding anti-derivatives.