Partial Derivatives
Stand on a hill whose height obeys `f(east, north)` and ask how steep the ground is; it answers with a question, steep which way? Face north, then face east, and the same spot gives two different answers.
- ▸One spot on two-input terrain has no single steepness: north climbs, east can fall away.
- ▸Freeze your east position and the hill collapses to one ordinary north-south curve whose slope you can already read.
- ▸Every knob a model turns gets its own steepness of the error, which is why each is nudged alone.
Every hill you have measured so far ran along a single line: one input in, one height out, and the steepness under your foot was a single number. Step onto real terrain and that breaks. The ground now rises with two things at once, how far east you have walked and how far north, and its height is a rule fed both. Stand at one spot and ask the old question, how steep is it here, and the ground refuses a single answer. Face north and the slope climbs hard; turn on the spot and face east and the same patch of dirt might fall away under you. Same footing, same instant, two contradictory steepnesses, because steepness on a surface is not a property of the spot, it is a property of the spot and the direction you chose to walk. Before any of the old machinery can run, that ambiguity has to be pinned down.