Today, I answered this question and discovered that the definition of $\dfrac{\partial^2}{\partial x\partial y}$ is a matter of convention.
For example this .edu link and this other .edu link use the convention $$\frac{\partial^2}{\partial x\partial y}:=\frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}\right) \qquad \qquad (1)$$
However, this wikipedia article, this .edu link and this other .edu link use the convention
$$\frac{\partial^2}{\partial x\partial y}:=\frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}\right) \qquad \qquad (2)$$
Since apparently the definition has not been fixed yet, I can imagine that both definitions has advantages/disadvantages depending on the context they are used. However, I can't find any of these situation.
Question: What are the pros and cons of each definition?
EDIT: At the time I write this edit, it seems that even in the wikipedia article about partial derivatives, there is a "contradiction", here definition $(1)$ is used but here they use definition $(2)$...
We all know that mixed derivatives are equal under very mild hypotheses, and whenever these hyptheses are unquestionably fulfilled nobody cares wether we should write ${\partial^2 f\over \partial x\partial y}$ or ${\partial^2 f\over \partial y\partial x}$.
As soon as there is only a hint of a suspicion that the order of differentiation could matter any author would precise on his first page what exactly is meant by ${\partial^2 f\over \partial x\partial y}$.
To sum it all up: This is a very tiny notational ambiguity left over in the sea of mathematics that should detract nobody from addressing the heart of the matter at hand.