Forward or Reverse: Filling the Jacobian

Here is a function from the plane into space: every point of the floor is lifted onto the surface above it. Its Jacobian, on the right, has three rows, one per output, and two columns, one per input.

Forward mode pushes an input direction through the function. Push the first axis direction, and it comes out as the teal arrow along the surface: that arrow is the first column of J. One J V P fills one column.

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