The interactive film is in the book's Companion

Play it, pause it, turn it and drag it with Mathematics for AI and Machine Learning.

Get the book

The Implicit Function Theorem

Chapter 11 · Mathematics for AI and Machine Learning · Advanced · 1:41

An interactive live film: the surface F(x, theta) = x^2 + theta^2 - 4 meets height zero in a circle; drag a point along it to see the slope dx*/dtheta = -(dF/dx)^(-1) dF/dtheta from two slices, and find where dF/dx = 0 and the theorem fails.

Narration

Here is a surface, F of x and theta equals x squared plus theta squared minus four, over the floor of theta and x. The light sheet marks height zero.

Where the surface crosses the sheet, F equals zero: the green circle, copied onto the floor, holds every pair where x star of theta solves the equation.

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

More from Chapter 11

Interactive1:54Backpropagation, One VJP per LayerChapter 11Preview1:27Partial Derivatives Are Two SlicesChapter 11Preview1:34The Gradient Is the Steepest DirectionChapter 11Preview1:42Jacobian Matrix: Linearizing a MapChapter 11Preview1:43The Derivative of the Matrix InverseChapter 11Preview1:54Forward or Reverse: Filling the JacobianChapter 11Preview1:32Taylor Expansion, One Term at a TimeChapter 11Preview1:30Second-Order Taylor: Where the Quadratic Model Can Be TrustedChapter 11Preview1:38A Hessian-Vector Product from Two GradientsChapter 11Preview1:34The Gauss–Newton MatrixChapter 11Preview1:44Hessian Matrix: Bowls, Saddles and Curvature DirectionsChapter 11Preview1:37Why the Hessian Is SymmetricChapter 11Preview1:26The Newton Step: Jumping to the Taylor Bowl's BottomChapter 11Preview2:36Corners, Maxima and Steps in the Chain RuleChapter 11Preview1:59GPTQ: Rounding with the HessianChapter 11Preview2:12AWQ: Activation-Aware Weight QuantizationChapter 11