Which is the Correct definition of $f_{xy}$

Question

I have seen some authors define $f_{xy}=\frac{\partial^2f}{\partial x\partial y}$

as:

$$f_{xy}=\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial y}\right)$$

and some authors define in an alternate way. Which is correct?

Answer 1

The alternate way is correct. The difference between these two traditional notations for partial derivatives is that the subscript notation is postfix and the Leibniz notation is prefix.

When we use the subscript notation, we write the letter denoting the variable of differentiation after the name of the function. That's why, for example, for a function $f(x,y)$ of two variables, its first derivative with respect to $x$ is $f_x$ and its first derivative with respect to $y$ is $f_y$. Therefore, $f_{xy}$ means

$f_{xy}=(f_x)_y:$ first with respect to $x$ and then with respect to $y$.

When we use the Leibniz notation, we write the differentiation operator before the name of the function. That's why, for example, for a function $f(x,y)$ of two variables, its first derivative with respect to $x$ is $\frac{\partial}{\partial x}f=\frac{\partial f}{\partial x}$ and its first derivative with respect to $y$ is $\frac{\partial}{\partial y}f=\frac{\partial f}{\partial y}$. Therefore, $\frac{\partial^2f}{\partial x\partial y}$ means

$\displaystyle\frac{\partial^2f}{\partial x\partial y}=\frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}f\right):$ first with respect to $y$ and then with respect to $x$.

As you can see these are two different things. The same things are the following: $$f_{xy}=\frac{\partial^2f}{\partial y\partial x} \quad \text{and} \quad f_{yx}=\frac{\partial^2f}{\partial x\partial y}.$$

Of course, there's also Clairaut's Theorem that says that mixed partials are equal when they are continuous. So when we have reasons to assume or know that they are equal, we get careless and write the partials notation in any order we want, and that applies to textbooks too. But this is only valid after Clairaut's Theorem has been established, not at all as a definition.