The nudge ratio, one dimension up
Chapter 2 built the derivative as a nudge ratio: move the input a little, watch the output move, and compare the two. With one input there was nothing to decide. Forward or back was the whole choice, and the ratio came out as a single number.
On a surface that stops being true the moment you nudge. A floor is a plane, so "a little that way" needs a that way, and a direction with a size is a vector. That is why chapter 4 stopped to install the vocabulary. This chapter spends it.
Two directions come with names: hold y still and slide along x, and the
rise is the partial derivative α; do it the other way and you get
β. Glue each run and rise together and you get a tangent vector standing on the
surface. The question the whole chapter is really asking is whether those two are enough for the
other three hundred and fifty-eight.
They are, and the reason is the one idea from part one. Every direction on the floor is a
combination of the two basic blocks, so its tangent vector is the same combination of the two
tangent vectors. The whole floor follows those two up to a single tilted plane, and the rise in any
direction reads off it as α cosθ + β sinθ. Written as a vector
of the two partials dotted with the direction, that is
Duf = ∇f · u. The gradient was never anything more exotic than
those two numbers in a column.
The last third spends the formula rather than proving more of it: why the steepest climb is along the gradient, why implicit differentiation is a level curve refusing to change altitude, and why crowded contour lines mean steep ground. The final line, two gradients lining up where two curves touch, is the one the economics series leans on hardest.
Chapters in the video
- 0:00What a two-variable function is
- 2:09The surface, and its contour lines
- 3:46A slope for every direction
- 5:28The derivative as a nudge ratio
- 6:35Gluing the nudges into a tangent vector
- 7:41Two partial derivatives, and the tangent plane
- 12:54The gradient, and what a dot product means
- 14:30Where implicit differentiation comes from
- 16:08Reading a contour map
- 17:55Part two in one breath
Read it instead
The written version:
implicit differentiation is a level curve's slope
— where dy/dx = −fx/fy comes from, and why the
minus sign was never arbitrary.
Walk inside it
The hillside shots in this chapter are recorded inside Open World Discover, and the ground you walk on there is the same surface as the one in the animation, fed straight into the terrain. Turning around really is sampling the slope in every direction.