# Doubt on one Maxwell relation

In thermodynamics, we always need to use exact differential relations to find the so called Maxwell relations. For a function of x and y, z=z(x,y)

if dz = M(x,y)dx + N(x, y)dy

the complete conditon for above equation to be hold is

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

Now, let see one of the relation on Helmholtz free energy

$$dF = -SdT - PdV$$

apply the condition mentioned above, we get

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

Well, in many materials (including some textbooks), they like to write it

$$\left ( {\partial T\over \partial p} \right )_{V,N} = \left ( {\partial V\over \partial S} \right )_{T,N}$$

I don't know why they like to inverse those relation! What interesting is only this relations I found to be in reverse order, the others Maxwells relations are just fine!

Mapes
There's absolutely no difference; the reciprocals of Maxwell relations are also valid. We can also get to this one by looking at the potential $H=E+PV$ rather than $F=E-TS$. It is odd that some textbooks would switch just one without comment. It would be a good teaching opportunity to explain that this is OK.