At a critical point of a surface, two forces compete. A and C bend it along the axes, trying to close it into a bowl. B, the mixed derivative, twists it into a saddle. One number decides who wins.
Drag A, B and C and watch the surface answer. Probe the curvature in any direction, then play the twist battle and see the bowl break the instant B² passes AC.
Narrated storyTen scenes on the mixed derivative fxy: stand on the surface, stick your arm out along x, step sideways in y — and feel the ground ahead of you tilt.