Partial Differentiation Proofs

The Total Derivative

Let the inductive hypothesis, P_n, be: for small displacements dx¹, ,…, dxⁿ
Suppose P_n is true for some n. Then, for small dx^n + 1, if second derivatives exist,
So P_n implies P_n + 1. But P₁ is true, so P_n is true for all n by induction. Hence,
Return

Clairaut’s Theorem

It is perhaps more surprising that there are second derivatives which don’t commute», than that it holds for well behaved functions. It is sufficient to prove the theorem in two dimensions, with variables x and y. I will use the notation:

If f_xy is bounded on the rectangle [a, b] × [c, d], then, by the fundamental theorem of the calculus,
Similarly,
Then
But if f_xy and f_yx are continuous and not equal at (x, y), there is a small rectangle [a, b] × [c, d] containing (x, y), such that
giving a contradiction.

Return