Curl is defined (in the plane) by imagining a wheel of radius $\epsilon$ placed in $(a,b)$. Denote the region enclosed by the boundary of the wheel with $D_\epsilon$. Let's suppose our vector field $\textbf{F}$ describes in every point in the plane some kind of velocity. If we integrate along the boundary $\partial D_\epsilon$ and then divide by the length of the curve comprising the boundary, it should result in some sort of average velocity about the boundary. So we can write
$$ v_{\text{avg.}} = \frac 1{2\pi \epsilon} \int_{\partial D_\epsilon}\!\!\!\! \textbf{F} \,{\scriptsize \bullet}\,\text{d}\textbf{r} $$
For circular motion, we know that $v = \omega r$ where $\omega$ and $r$ is angular velocity and radius of the circle respectively. In our case $v = v_{\text{avg.}}$ and $r = \epsilon$; therefore, by means of Green's theorem, we can write
$$ \omega = \frac 1{2\pi \epsilon^2} \int_{\partial D_\epsilon}\!\!\!\! \textbf{F} \,{\scriptsize \bullet}\,\text{d}\textbf{r} = \frac 1{2\pi \epsilon^2} \int_{\partial D_\epsilon} \!\!\!\!\! \text{Pd}x + \text{Qd}y = \frac 1{2\pi \epsilon^2} \iint_{D_\epsilon} \left( \frac {\partial \text{Q}}{\partial x} - \frac {\partial \text{P}}{\partial y} \right)\text{d}x\text{d}y $$
By using the mean value theorem for double integrals and letting $\epsilon \to 0$ we can then easily show that $$\omega = \frac 12 \left( \frac {\partial \text{Q}}{\partial x} - \frac {\partial \text{P}}{\partial y} \right)$$
And this is why we understand the integrand as the tendency to cause rotation on an infinitely small circle in $(a,b)$ (intuition derived from the known relationship $v = \omega r$ for circular motion)
Divergence is constructed in a similar fashion but in this case there is no obvious (physical) relationship I can think of as a vantage point. We want to look at the perpendicular components of the vector field along the boundary and I am not sure where to start off.
The final steps of the proving the divergence on an infinitely small circle in $(a,b)$ where we let $\epsilon \to 0$ is almost exactly identical to that of curl, that's not the issue at hand. I'd like to start off from a geometric point of view but I am not sure how to, any thoughts?