I just recently learnt the Maxwell Relations in Thermodynamics. We aren't really doing anything with them, just went through the derivations. In deriving them, we started with the equation of state: TdS=dU+PdV where T is temperature, S entropy, U internal energy, P pressure, V volume. We would than pick two of the four variables S,P,V,T, and with some math derive one of the maxwell relations. This was done using differentials, ie, if df(x,y)=Adx+Bdy, than A=df/dx (partial derivative) and B=df/dy (p.d.), also, as long as the function isn't pathological, dA/dy=dB/dx (p.d.). Here's my question, each of the four maxwell relations is found by chosing 2 of the four variables, and there are four maxwell relations. But there are 6 possible ways to chose 2 of the 4 variables, so how come there aren't 6 Maxwell Relations? Is there simply no way to manipulate the resulting equations to make it work for the other 2 pairs of variables, or is there some other reason for this? Thank you.