As an interesting aside, the n-th partial derivatives form an n-cube:

For m variables, there are m first orders, m^2 second orders, etc. and most proofs about, eg convexity, amount to a proof over that shape.

The reason that n-th order reasoning about an m-dimensional problem is difficult is that it requires reasoning using m^n objects.

If you figure we can remember about ten things intuitively, that’s just enough for three dimensions and two derivatives — enough to do motion analysis.

But it’s leagues away from being able to do the second order of a ten-dimensional problem intuitively.