Notation in thermodynamics derivatives In Yung Kuo  Lim's book of exercises in thermodynamics and Stat. Physics I have found more than once the following notation for partial derivatives (ex. 1081 page 79):
$$ \left(\frac{\partial T}{\partial L}\right)_S = \frac{\partial(T,S)}{\partial(L,S)} $$
I don't think that this is only a matter of notation, because the authors use it as a way to solve the exercises, not just as a different way to write something.
What does the writing on the RHS of the equation stand for?
EDIT: The author uses both notations, that's why I wondered if the second one has something more to say than the first one. Here's another example of how he uses it:
$$ \left( \frac{\partial T}{\partial H} \right)_S = \frac{\partial(T,S)}{\partial(H,S)} = \frac{\partial(H,T)}{\partial (H,S)} \cdot \frac{\partial(H,M)}{\partial(H,T)} \cdot \frac{\partial(T,S)}{\partial(H,M)} $$
with T the temperature, S the entropy, H the magnetic field and M the magnetization. If it is only a matter of notation, how is it the second one more useful than the first one?
 A: I think this is the answer: $\frac{\partial(y,x)}{\partial (z,w)}\equiv -\frac{\partial y}{\partial x}\Bigr |_{all~other~ variables~ constant}\Bigr/\frac{\partial z}{\partial w}\Bigr |_{all~other~ variables~ constant}$. I will explain below why minus sign is required. Whether or not my answer is right, a book with such crappy notation that it requires considerable effort to understand what the notation means is not worth spending your time over. I would recommend Thermodynamics by Callen. In particular read the chapter on Maxwell relations.
I make use of the identity: $\frac{\partial y}{\partial x}\Bigr |_z=-\frac{\frac{\partial y}{\partial z}\Bigr |_x}{\frac{\partial x}{\partial z}\Bigr |_y}$. This is where the minus sign comes in.
If my definition is correct then:
$\frac{\partial T}{\partial H}\Bigr |_{S,M}=-\frac{\frac{\partial T}{\partial S}\Bigr |_{H,M}}{\frac{\partial H}{\partial S}\Bigr |_{T,M}}=\frac{\partial (T,S)}{\partial (H,S)}$
Taking the expression on the extreme right:
$\frac{\partial (H,T)}{\partial (H,S)}\frac{\partial (H,M)}{\partial (H,T)}\frac{\partial (T,S)}{\partial (H,M)}=-\frac{\frac{\partial H}{\partial T}}{\frac{\partial H}{\partial S}}  \frac{\frac{\partial H}{\partial M}}{\frac{\partial H}{\partial T}}  \frac{\frac{\partial T}{\partial S}}{\frac{\partial H}{\partial M}}=-\frac{\frac{\partial T}{\partial S}}{\frac{\partial H}{\partial S}}=\frac{\partial T}{\partial H}\Bigr |_{S,M}$
where in the intermediate steps I have suppressed names of variables which are held constant during partial differentiation to avoid clumsiness of appearance.
A: The notation stands for the following:
$$
\partial \left( \left[\text{variable  to  differentiate} \right], \left[\text{variable to hold  constant} \right] \right)
$$
As @PeterShor pointed out, a multi-variate system requires one to worry about which variables to hold constant during partial differentiation.
You could equivalently write this as:
$$
\left( \frac{ \partial \left[\text{variable to differentiate} \right] }{ \partial \left[ \text{variable of differentiation} \right] } \right)_{\left[\text{variable to hold constant} \right]}
$$
as there are multiple forms of notation.
Note that partial derivatives require you to specify with respect to which variables to hold constant but total derivatives differentiate every variable of which the differentiated variable is a function.
Side Note:  There is a specific name for the variable in the denomenator, which I am not recalling off hand.  Perhaps a more mathematically inclined individual could help?
A: I have found the answer to my own question in Callen, Thermodynamics and Thermo-statistics:
\begin{equation}
\frac{\partial(u,v,\ldots,w)}{\partial(x,y,\ldots,z)} \equiv \det \left(\matrix{\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} &\ldots &\frac{\partial u}{\partial z}\\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} &\ldots &\frac{\partial v}{\partial z}\\
\vdots &\vdots &\ddots &\vdots\\
\frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} &\ldots &\frac{\partial w}{\partial z}\\}\right),
\end{equation}
which is just a Jacobian matrix.
The following identities hold:
\begin{align}
\frac{\partial(u,v,\ldots,w)}{\partial(x,y,\ldots,z)} &= -~\frac{\partial(v,u,\ldots,w)}{\partial(x,y,\ldots,z)}\\
\frac{\partial(u,v,\ldots,w)}{\partial(x,y,\ldots,z)} &=\frac{\partial(u,v,\ldots,w)}{\partial(r,s,\ldots,t)} \frac{\partial(r,s,\ldots,t)}{\partial(x,y,\ldots,z)}\\
\frac{\partial(u,v,\ldots,w)}{\partial(x,y,\ldots,z)} &=1\left/\frac{\partial(x,y,\ldots,z)}{\partial(u,v,\ldots,w)}\right..
\end{align}
The link with thermodynamic relations comes from the identity:
\begin{equation}
\left(\frac{\partial u}{\partial x}\right)_{y,\ldots,z} = \frac{\partial(u,y,\ldots,z)}{\partial(x,y,\ldots,z)},
\end{equation}
hence the relation of the example in the question.
