A Hessian-Vector Product from Two Gradients

Here is a loss surface in two dimensions, a point x, and a direction d. The blue arrow is the gradient at x. Now nudge x by epsilon along d, and draw the gradient there: the teal arrow.

The change in the gradient, divided by epsilon, is the amber arrow. As epsilon shrinks, it settles onto H times d, the Hessian vector product, drawn as the dashed arrow. We never formed H.

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