Working in the xy-plane (ie in an attempt to work in 2D) the curl equation becomes
I set out to work out what this actually means. I drew a vector field of parallel vectors heading up from bottom left to top right and increasing in magnitude in a uniform manner.
In the bottom right hand corner of the picture above I have shown that each vector
F made up of an x-component Fx and a y component Fy. On the diagram below I have shown how the size of those y component changes as you head out along the x-axis and how the size of the x component changes as you head up the y-axis. The first term in the top equation tells you the rate at which the y component increases as you go along the x-axis. The second term is the rate of increase of the size of the x component with increase in y. If the x and y components increase at the same rate, the vectors will continue to point along their straight slanted line. The equation will give an overall value of zero. If the y component increases at a faster rate that the x component, the arrows would tend to become more vertical as they go from bottom left to top right. The line of vectors will twist and the top equation will give a positive value. The y components twisting in an anticlockwise direction will tend to beat the clockwise x components. The twist will be anticlockwise. It fits with the right hand grip rule. I think this is the meaning of the curl equation.
