Why are some Maxwell relations written in reverse order?

[KFC]

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.

 

[jojo12345]

this page: http://en.wikipedia.org/wiki/Exact_differential#Reciprocity_Relation
might be enlightening. I know it helped me. At the school I go to, most of the engineers or physics majors (same math requirement more or less) I have talked to don't really know about these rules. I'm pretty sure that means we were never taught them. It's kind of annoying.