TITLE: Notation for partial derivatives
QUESTION [37 upvotes]: I thought that the meaning of
$$
\frac{\partial f(x, y, z)}{\partial x}
$$
is differentiation on $x$ with fixed $y$ and $z$. So $(x, y, z)$ in the numerator is just saying which variables are fixed. If I need to indicate where the derivative is evaluated, I write it in the right of a vertical bar as a subscript. But today my teacher used $(x, y, z)$ in the numerator to denote where the derivative is evaluated. So, for example,
$$
\frac{\partial f(0, 0, 0)}{\partial x}
$$
means
$$
\frac{\partial f(x, y, z)}{\partial x} \bigg\rvert_{x=0,y=0,z=0}
$$
Is that a standard convention? If so, what is the meaning of
this?
$$
\frac{\partial f(x, y, g(x, y))}{\partial x}
$$
I have two candidates. One is a partial derivative of the composition of $f$ and $g$ where $g$ has some fixed value, and the other is the partial derivative of $f$ on $x$ evaluated at $(x, y, g(x, y))$. I think the two are not the same.

REPLY [2 votes]: I find the answer of user21820 particularly nice. But a few remarks on the historical origin of this "notational mess" might be enlightening.
Jacobi popularised the partial derivatives notation   with his 1840 paper De determinantibus functionalibus, where he spends 3 pages introducing the notation, explaining a certain problem he sees with it, and suggesting a solution for this notational problem. Surprisingly, the problem Jacobi saw was not the one the OP sees. Instead, it was probably Jacobi's suggested solution which leads to the  problems the OP an everyone else in this thread sees! 
You may read what Jacobi he had to say here. But beware that in order to understand what he was saying (and lead any reasonable discussion about it), one needs to understand the difference between how the word function was used prior to $\approx$ 1930, and how we officially use it today. I've written about this distinction here and here.
In case you don't want to read Jacobi, here's a short rephrasing of Jacobis problem by Vladimir Arnold in his Classical Mechanics book p. 258 footnote 81:

It is necessary to use the apparatus of partial derivatives, in which
  even the notation is ambiguous$.^{81}$

It is important to note that the quantity $\partial{u}/\partial x$ on the $x, y$-plane depends not only on the function which is taken
  for $x$, but also on the choice of the function $y$: in new variables
  $(x, z)$ the value of $\partial{u}/\partial x$ will be different. One
  should write $$\left.\frac{\partial u}{\partial x}\right|_{y=\text{const.}} \quad \left.\frac{\partial u}{\partial x}\right|_{z=\text{const.}}$$


In the thermodynamics literature the same problem is solved with the notation 
$$\left(\frac{\partial u}{\partial x}\right)_{x,y} \quad \left(\frac{\partial u}{\partial x}\right)_{x,z}$$
but Jacobi's original suggested solution was writing
$$\frac{\partial u (x,y)}{\partial x} \quad \frac{\partial u (x,z)}{\partial x}$$
(!) Combine this with his awkward choice of the letter $f$ for what  Arnold denotes with $u$ and you have guaranteed confusion. (I very much suspect that it was also Jacobi who opened the gate for the awfully popular $y=y(x)$).