I saw the following theorem in class.
Theorem (Young): Let $f:\mathbb{R}^2\to\mathbb{R}$ partially differentiable with partial derivatives both differentiable at $x_0\in\mathbb{R}^2$. Then $f_{xy}(x_0)=f_{yx}(x_0)$.
Next the professor mentioned the following.
Theorem (Schwarz): Let $f:\mathbb{R}^2\to\mathbb{R}$ partially differentiable and such that $f_x$ is $y$-differentiable and $f_{xy}$ is continuous at $x_0\in\mathbb{R}$. Then $f_y$ is $x$-differentiable at $x_0$ and $f_{yx}(x_0)=f_{xy}(x_0)$.
Note: I'm using $f_{xy}=(f_x)_y$.
Now, in the notes, the second theorem is actually formulated in Spanish in a way that I can't tell if the theorem requires only $f_{xy}$ to be continuous at $x_0$ or if also $f_x$ and $f_y$ should. This led me to search the actual statement of the theorem online and what I found is this
Theorem (Schwarz): Let $f:\mathbb{R}^2\to\mathbb{R}$ of class $C^2$. Then $f_{xy}=f_{yx}$.
which is a particular case of (what supposedly is, but frankly now I don't know who to trust) Young's theorem! So there are two questions I need an answer to.
- What's the correct statement of Schwarz's theorem?
- If my Schwarz's theorem is not Schwarz's theorem, then whose is it and what is its correct statement?
Note: I know the theorems are all stated for more than two dimensions. But since the proof relies on the 2D case, we prefer to just enounce for that case. It's also easier to state and to visualise. My doubt is in no way related to the number of dimensions but to the other conditions of the theorem.
According to a couple of sources I kind of trust, Scharwtz' theorem says that if some function $f(x_1, x_2, \cdots ,x_n)$ has continuous second derivatives $\frac{\partial f}{\partial x_i \partial x_j}$ for all combinations $i,j$ in some open ball around some point $X$, then those second derivatives commute, in the sense that $\frac{\partial f}{\partial x_i \partial x_j} = \frac{\partial f}{\partial x_j \partial x_i}$.
Note that this applies to functions of two or more variables.
Now, on Wikipedia (https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives) it states that Young's theorem is the same as Schwartz' theorem. I had only seen Young's theorem, when discussed in isolation, stated as applying to just two variable functions. So, that might have been a difference. But the first reference in that wikipedia article is to a paper showing that it did refer to arbitrary numbers of variables.
One would have to go back to original source material to see if Schwartz stated his theorem only for two variables; but it is often the case that mathematicians name a theorem after a key worker who never actually stated the theorem in the form used at that later time, so this issue might not have an easy resolution.