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

My book doesn’t give any kind of explanation of it. Again, I’m

notlooking 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 ∂∂y∂∂xf, we have

fxy≈1h(fx(D)−fx(A))≈1h2((f(C)−f(D))−(f(B)−f(A))).

Similarly,

fyx≈1h2((f(C)−f(B))−(f(D)−f(A))).

But those two things are the same: they both correspond to the “stencil”

1h2[−1+1+1−1].

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