# Condition for equality of mixed derivatives

It says that theorems 12.11 and 12.12 imply Theorem 12.13. But, don't we need some extra conditions? Like existence of $D_{r,r}f$ and $D_{k,k}f$? Here $f$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $D_kf$ denotes the partial derivative of $f$ w.r.t. the $k^{th}$ variable.

The images are from Tom Apostol's Mathematical Analysis.

• To say that $D_{k.r}(c)$ is continuous implies that it exists.
– san
Aug 24, 2015 at 2:05
• Baby Rudin has a stronger version ( in the sense that it is based on weaker condition). See theorem 9.41.
– Vim
Aug 28, 2015 at 9:44
• Also asked (but not answered) two weeks earlier by another user: math.stackexchange.com/questions/1383105/… Jan 1, 2020 at 10:10

## 2 Answers

I think that the confusion is in perhaps too much detail. Theorem 12.11 states the "existence" of derivatives in c and the continuity of other of other derivatives in n-ball, is a sufficient condition, for making f differentiable.

Theorem 12.12 states, the existence of both partial derivatives and and both are differentiable at c, then the relation is exact, and commutable. (Very interesting result that gets used extensively in Electromagnetic/Quantum Fields... but that's the digression.)

Theorem 12.13, which adds "continuity" in Theorem 12.12, flows straight from the 12.12, in my view, there never was the separate need for 12.13, as the "continuity" could've been included/stated in the 12.12, clearing up the verbiage a bit, perhaps.

And the stated conditions are "sufficient" to determine the "uniqueness", please read the condition in their geometric perspective to ascertain the uniqueness of the conditions, implied as to sufficiently adjudicate the behaviour of f, hence their ability to ascertain the differentiability and continuity, and the "sufficient-ness" of this ability.

• A function that satisfies the conditions of Theorem 12.13 need not satisfy the conditions of Theorem 12.12. So, how can we proceed in such a case? Do the conditions of 12.13 imply those of 12.12? Aug 28, 2015 at 8:28

The condition that $D_{r,k} f, D_{k,r}$ are continuous in an n-ball $B(c)$ implies that $D_{r,r}, D_{k,k}$ exit.

• How does one prove that? Aug 28, 2015 at 16:46