How to understand curl means rotation

Question

Just by looking at the definition of curl:

$${\displaystyle \nabla \times \mathbf {F} =\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}+\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}}$$

how to understand this measures the magnitude of circulation?

use the X axis as example:

$$\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}$$

It is the rate of change of z component of F with respect to change of y minus the rate of change of y component of F with respect to change of z. how is that related to rotation?

Thanks!

Relevant thread about how to discover the classical Stokes’s theorem: https://math.stackexchange.com/q/4323512/40119 I compute the circulation around a tiny parallelogram and the curl formula pops out. — littleO, Oct 08 '23 at 01:29
This may be of interest: https://www.youtube.com/watch?v=rB83DpBJQsE — user10478, Oct 08 '23 at 01:38
Look at the case $\mathbf{F}=-y\mathbf{e}x+x\mathbf{e}_y=\rho\mathbf{e}\theta$ where $(\rho,\theta,z)$ is the cylindrical coordinates. — user10354138, Oct 08 '23 at 01:59
I thought the answer that @Justauser posted then deleted had some useful intuition. Maybe they will undelete it. — littleO, Oct 08 '23 at 21:22

How to understand curl means rotation

0 Answers0