I am really not sure whether this question can be posted in MSE or not, but hopefully can be. If not, kindly do not downvote, but suggest me to delete it from here and post it somewhere else.
While reading an old scanned book (not clear/blurred), I have found this multiple choice question:
Which, using Latex, is:
If $u=f(x,y)$ and the partial derivatives $\frac{\partial u}{\partial x}$ and $\frac{\partial y}{\partial x}$ are themselves differentiable functions of $x$ and $y$, there partial derivatives are denoted by:
I. $\frac{\partial^2 u}{\partial x^2}$
II. $\frac{\partial^2 u}{\partial y \partial x}$
III. $\frac{\partial^2 u}{\partial x \partial y}$
IV. $\frac{\partial^2 u}{\partial y^2}$
Which of the above are always equal?
(A) I and II
(B) II and III
(C) III and IV
(D) I and III
(E) II and IV
The solution is:
Which, using Latex, is:
(B)
If $u=f(x,y)$, and if $\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}$ and $\frac{\partial^2 u}{\partial x \partial y}$ are continuous, then $\frac{\partial^2 u}{\partial x \partial y}=\frac{\partial^2 u}{\partial y \partial x}$; that is, the order of differentiation is immaterial.
Now, my question is; in the problem statement, "If $u=f(x,y)$ and the partial derivatives $\frac{\partial u}{\partial x}$ and $\color{red}{\frac{\partial y}{\partial x}}$ are themselves ...", should not it be "If $u=f(x,y)$ and the partial derivatives $\frac{\partial u}{\partial x}$ and $\color{green}{\frac{\partial u}{\partial y}}$ are themselves ..."?
I have included the solution so one may find if this is typo or it is okay.
Your help would be appreciated. THANKS.