The interactive film is in the book's Companion

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Why the Hessian Is Symmetric

Chapter 11 · Mathematics for AI and Machine Learning · Advanced · 1:37

An interactive live film: the book's proof of Schwarz's theorem on a small square lifted onto a twisted surface; read the twist along x0 or along x1, it is the same number, and it tends to the mixed partial as h shrinks.

Narration

Here is a twisted surface and a point x on it. Schwarz's theorem says that the two mixed second partial derivatives agree, so the Hessian is symmetric.

Take a small square of side h at x, and lift its four corners onto the surface.

The rest of the narration is in the book's Companion.

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