# Notation of Maxwell relations

The Maxwell relations are often given as for example

$$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V.$$

What does the $S$ and the $V$ in the index of the parantheses mean? I guess that $S$ and $V$ should stay constant for the derivation, but is this not already in the definition of the partial derivative?

• en.wikipedia.org/wiki/Maxwell_relations Best place to solve this doubt. – Yuzuriha Inori Mar 12 '18 at 9:55
• It does not explain the notation, does it? – Alduno Mar 12 '18 at 9:59
• The derivation part makes it pretty clear as to what the variables mean and what the notations are – Yuzuriha Inori Mar 12 '18 at 10:01
• There are many more than two parameters used in thermodynamics. This just makes clear exactly which ones are being held constant. – Chet Miller Mar 12 '18 at 12:53

Basically in Thermodynamics different functions get the same name if they refer to the same quantity. So, for example, the inner energy $U(p,V,N)$ and $U(T,V,N)$ both are called $U$ although they are not the same function.
Your system has two degrees of freedom (three if you include $N$, the "number of particles", but that's not relevant here). So any of your quantities $V$, $E$, $P$, $T$, $S$ can be viewed as a function of any two of the others. The expression $$\left(\frac{\partial T}{\partial V}\right)_S$$ means "the derivative of $T$ with respect to $V$ when viewing it as a function of $V$ and $S$ (i.e. $T(V,S)$)". Likewise $$\left(\frac{\partial P}{\partial S}\right)_V$$ is the derivative of $P(V,S)$ with respect to $S$.