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 f_{x}
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 f_{y}. Now if we move the pen
in the x-direction, we measure f_{yx}. The video to the right
illustrates this: