Q:

Proof of the definition corollary of curl

$$(\nabla \times \mathbf{F}) \cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{A \to 0}\left( \frac{1}{|A|}\oint_{C} \mathbf{F} \cdot d\mathbf{r}\right)$$
This is how the curl operator is usually defined, and I want to know the proof that the left hand and the right hand are equivalent.

A:

If A is an element of surface area in bounded by a simple closed curve C, let P be an interior point in C and $\mathbf{n}$ a unit normal at P. 
By Stokes' theorem, $\iint(\nabla\times \mathbf{F})\cdot \mathbf{n}~dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r}.$
Using the mean value theorem for integrals* we can write this as 
mean[$(\nabla \times \mathbf{F} )\cdot \mathbf{n}$] =$ \frac{\oint_{C} \mathbf{F} \cdot d\mathbf{r}}{\Delta A},$ and the result follows from taking the limit as $\Delta A \to 0. $
In words, the expression for (curl$\cdot\mathbf{n}$) reaches a limiting value as the area A shrinks around the point P.
*The MVT for integrals is:
If a function f is continuous on [a,b] there exists a point c on [a,b] such that $$f(c) = \frac{1}{b-a}\int_a^b f(x)dx.$$ 

