I’m looking for a geometric interpretation of this theorem:
My book doesn’t give any kind of explanation of it. Again, I’m not looking for a proof – I’m looking for a geometric interpretation.
Inspired by Ted Shifrin’s comment, here’s an attempt at an intuitive viewpoint. I’m not sure how much this counts as a “geometric interpretation”.
Consider a tiny square ABCD of side length h, with AB along the x-axis and AD along the y-axis.
D---C | | h A---B h
Then fx(A) is approximately 1h(f(B)−f(A)), and fx(D) is approximately 1h(f(C)−f(D)). So, assuming by fxy we mean ∂∂y∂∂xf, we have
But those two things are the same: they both correspond to the “stencil”