Intuition behind $\nabla \times \mathbf{F}$ Is there a simple explanation why this form for the curl of a vector field $\mathbf{F}$,
$$\nabla \times \mathbf{F}=\begin{vmatrix}
\hat{x} & \hat{y}  &\hat{z}  \\ 
 \frac{\partial}{\partial x}& \frac{\partial}{\partial y} &\frac{\partial}{\partial z} \\ 
 F_x& F_y &F_z 
\end{vmatrix}$$
Corresponds to the amount of 'twiting' of $\mathbf{F}$ (and any other qulities of $\nabla \times \mathbf{F}$)?
When I first saw the equation, it seemed, very roughly, to be a measure of how much a component of $\mathbf{F}$ is affected by the other two components. However, this only really differentiates between $0$ and ' not $0$' curl, and anyway  there are thousands of possible equations that would give the same first impression. What's so unique about this one?
 A: This is completely explained by Stokes' theorem:
$$\oint_{\partial S} \vec{F} \cdot d\vec{r} = \iint_S (\nabla\times\vec{F}) \cdot d\vec{n},$$
where $d\vec{n}$ is the infinitesimal normal to the surface $S$ and $d\vec{r}$ is the infinitesimal tangent to its boundary $\partial S$, oriented "positively" (according to the right-hand rule).  If you take $S$ to be an infinitesimal disk of radius $\epsilon$ oriented perpendicular to a fixed vector $\vec{u}$ and centered at some $\vec{v}$, then you get
$$\iint_S (\nabla \times \vec{F}) \cdot d\vec{n} \approx \pi\epsilon^2 (\nabla \times \vec{F})(\vec{v}) \cdot \vec{u}$$
while the line integral around $\partial S$ is interpreted, physics-ly, as the work done by $\vec{F}$ around the circular boundary.  Therefore, dividing and taking a limit, you get
$$(\nabla \times \vec{F})(\vec{v}) \cdot \vec{u} = \lim_{\epsilon \to 0} \frac{1}{\pi \epsilon^2} \int_{\partial S} \vec{F} \cdot d\vec{r},$$
meaning that the component of the curl of $\vec{F}$ along a particular direction given by $\vec{u}$ is (basically) the work done by $\vec{F}$ while moving in a small circle around the $\vec{u}$ axis.
This is likely explained in most multivariable calculus textbooks; I know it's in Stewart, from which I lifted this almost verbatim from my memory of the class I taught from it last semester.
