Geometric intuition behind curl of vector field If F = P i + Q j + R k is a vector field on $\mathbb{R}^{3}$, then the curl of F is defined by 
$$\operatorname{Curl}(F) = \nabla \times F$$
where $\nabla$ is the differential operator.
Is there a geometric intuition behind the curl of a vector function F? I would like to see a clear geometric diagram explaining the concept of the curl of F, as my book only gives the definition.
Any help is appreciated!
 A: I have a good picture for a 2-dimensional vector field, which can be adapted for 3 dimensions. For the curl at a point in $\mathbb{R}^2$, I imagine placing a little flywheel at that point, so that it can be made to spin by the vectors around it. If you have a field given by, say $F(x,y) = \langle 0,x\rangle$ then a flywheel sitting at $(1,0)$ would have a little more upward push on its right side than on its left side, so it would rotate counterclockwise. On the other hand, with $F(x,y)=\langle 0,1\rangle$, the forces would be equal on either side, so the wheel would remain stationary.
Now, in $\mathbb{R}^3$, imagine locating that flywheel somehow in space so it is free to move itself to whatever orientation results in the fastest spin. Alternatively, let the flywheel be a ping pong ball, staying in its location but free to spin, and let the vector field be wind. Which way would the ping pong ball spin, and how quickly?
A: As @G-Tony-Jacobs, I will mostly restrict the discussion to dimension $2$. Let us assume $\vec{F}$ is of the form:
$$
\vec{F}(x,y, z) = \vec{F}(x,y) = F_1(x,y) \vec{i} + F_2(x,y) \vec{j} + 0 \vec{k}.
$$ Then, we have
$$
\operatorname{curl} \vec{F} = \bigg( 0,0, \frac{\partial F_2}{\partial  x} - \frac{\partial F_1}{\partial y} \bigg).
$$
We can define a circulation density as
$$
\text{circulation density } = \lim \frac{\text{flux around a small rectangle}}{\text{area}},
$$ where the limit is taken as the area goes to zero.

We have
$\bullet$ flux in the direction of $S_1 \simeq \big( \vec{F}(x+\Delta x, y) \cdot \vec{j} \big) \Delta y = F_2(x+\Delta x, y) \Delta y$
$\bullet$ flux in the direction of $S_2 \simeq - \big( \vec{F}(x, y+\Delta y) \cdot \vec{i} \big) \Delta x = - F_1(x, y+\Delta y)\Delta x$
$\bullet$ flux in the direction of $S_3 \simeq - \big( \vec{F}(x, y) \cdot \vec{j} \big) \Delta y = - F_2(x, y)\Delta y$
$\bullet$ flux in the direction of $S_4 \simeq \big( \vec{F}(x, y) \cdot \vec{i} \big) \Delta x = F_1(x, y) \Delta x$
Thus,
$$
\begin{split}
\text{circulation density at } (x,y) & = \lim_{\Delta x, \Delta y \to 0} \frac{F_2(x+\Delta x, y) \Delta y - F_1(x, y+\Delta y)\Delta x - F_2(x, y)\Delta y + F_1(x, y) \Delta x}{\Delta x \Delta y} \\
       & = \lim_{\Delta x \to 0} \frac{F_2(x+\Delta x, y) - F_2(x, y)}{\Delta x} + \lim_{\Delta y \to 0} \frac{F_1(x, y+\Delta y) -  F_1(x, y)}{\Delta y} \\
       & = \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y},
\end{split}
$$
More generally (and much in the same way), we can think that, for any vector field $\vec{F}$:
$\bullet$ The direction of $\operatorname{curl} \vec{F}$ is the rotation axis and the circulation obeys the right hand rule.
$\bullet$ A vector field is irrotational (borrowed terminology from fluid dynamics) when $\operatorname{curl} \vec{F} = \vec{0}$.
