Clairaut revisited

The visualization shown here from 2014 can be done simply by moving a pen over your knuckles. The critical point between two knuckles is a saddle point. If the pen is parallel to the x-axes, then we measure fx Now move the pen in the y direction. The slope changes. We can read off xy. Now do the same thing but having the pen parallel to the y-axes where the slope is fy. Now if we move the pen in the x-direction, we measure fyx. The video to the right illustrates this:

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