The Gradient Is the Steepest Direction

Here is the surface, f equals x zero squared plus x one squared minus x zero times x one, with its contour lines on the floor and a point x on it.

Pick a direction d, the black arrow on the floor. The wall along d cuts the surface, and the slope of that cut at our point is the directional derivative: the gradient dotted with d.

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