Partial derivatives commute. This is a basic fact about functions
of several variables, and anyone who's studied them knows it and has
seen it proved using a bit of algebra and some limits. But why
is it true?
Let z = f(x,y) be a function of two variables. In the diagram, a
small piece of the x,y plane is shown. I've assumed f(0,0) is
zero to keep the algebra simple.
The proof is contained in the diagram, and can be understood just by
looking at it. The following discussion is intended to help with
understanding the details of the diagram, but doesn't really add
As we move to the right by δx, the value of f, shown on the red line,
goes up by
If we then move to the rear by δy, the value of f increases by
we arrive at the blue dot, labeled
The difference between the blue dot and f(δx, δy) in that case is due
to the difference between the actual
partial of f with respect
to y at (δx,0) and the value we used, which was the partial of f with
respect to y at (0,0).
This difference is
On the other hand, if we go around the box the other way, then the
function first increases by
move to the rear, and then by approximately
move to the right. Again, we arrive at the blue dot:
And again, the actual value of f(δx,δy) will be different from the blue
dot -- but this time the difference will be due to the change in the
partial of f with respect to x
rather than y. The partial
of f with respect to x at (0,δy) will be different from its value at
(0,0), and the difference in the value of f will be
But the value of f(δx,δy) must be the same no matter how we get there,
so the second partials must be the same, no matter which order we
evaluate them in.
As with most "proof-by-picture" demonstrations, the point of this is
not rigor, but rather to show intuitively what's going on. We've
ignored a number of details here, such as the difference between the
second derivatives at the rear corner and the front corner. To
produce a rigorous proof of this requires explicitly evaluating the
limits used to define the derivatives to be sure that nothing we
ignored made a difference, and the algebra gets a bit messy. But
any standard calculus text should include a full proof of this.