Geometric interpretation of mixed partial derivatives?

I’m looking for a geometric interpretation of this theorem:

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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.

Thanks.

Answer

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 yxf, we have
fxy1h(fx(D)fx(A))1h2((f(C)f(D))(f(B)f(A))).
Similarly,
fyx1h2((f(C)f(B))(f(D)f(A))).
But those two things are the same: they both correspond to the “stencil”
1h2[1+1+11].

Attribution
Source : Link , Question Author : dfg , Answer Author : Community

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