Notation in partial differentiation

Question

What do we mean when we write $$\Bigg(\frac{\partial f}{\partial y}\Bigg)_x $$ should $x$ in $f$ be considered constant every time after we take $\Big(\frac{\partial f}{\partial y}\Big)_x $?
So in particular is $$\frac{\partial}{\partial x}\Bigg(\frac{\partial f}{\partial y}\Bigg)_x =? \ 0 $$

Answer 1

This notation is pretty common in thermodynamics. It just means that the derivative of $f$ is taken with respect to $y$ keeping $x$ constant i.e., it is just a normal partial derivative of $f$ with respect to $y$. The resulting entity is both a function of $x$ and $y$.

$$f=f(x,y) \\ x=x(y,...)$$

$(\frac{\partial f}{\partial y})_x$ is the derivative of $f$ with respect to $y$, where the subscript $x$ emphasises the fact that even though $x$ is a function of $y$ (or the other way around), it behaves like a constant while we take the derivative of $f$. An example is the following from Callen's Thermodynamics and an Introduction to Thermostatistics:

$S=S(U,V,N)$, where $S$ is entropy, $U$ is internal energy, $V$ is volume and $N$ is some other parameter. Here $U$ itself is dependent on $V$ and $N$, but the derivative of $S$ with respect to $U$ is taken considering all the $V$'s and $N$'s appearing in $S$ as constants. This gives the inverse of Temperature ($T$) as a function of $U$, $V$, and $N$. $$\frac{1}{T}=(\frac{S}{U})_{V,N}$$

In this convention $\frac{\partial}{\partial x}(\frac{\partial f}{\partial y})_x\neq 0$ in general.

Answer 2

$\left(\frac{\partial f}{\partial y} \right)_x = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y} \right)$. It is similar to that $\frac{\partial f}{\partial x}=f_x$.

Answer 3

The notation $$ \left(\frac{\partial f}{\partial y}\right)_x \stackrel{\text{def}}{=} \frac{\partial f}{\partial y} $$ and $x$ is not to be considered constant from thereafter. The use of this notation is to make it explicit which variables are being held constant. I think it is somewhat contrived, but it is a notation that is used commonly in statistical mechanics, for instance.

Note that this notation also clashes with the more common $f_x$ notation, which is to represent the partial derivative of $f$ with respect to $x$. These two notations are saying different things. One is saying, "these are all the variables that we are treating as constant when we evaluate the partial derivative" while the $f_x$ notation says "this is the partial with respect to $x$".

So, $$ \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)_x \stackrel{\text{def}}{=} \frac{\partial^2 f}{\partial x\partial y} $$ is not necessarily zero.