Hessian Matrix: Bowls, Saddles and Curvature Directions

Here is a landscape with both a valley and a pass: f equals x zero cubed minus three times x zero plus x one squared, and a point a on the floor.

Start at one and zero, on the right. Both principal curvatures are positive: six along the red curve, and two along the green curve. The local quadratic approximation, the blue surface, is a bowl curving upward in every direction.

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