As we have seen, the fundamental thermodynamic relation  implies that the natural variable in which to express  are  and : .

That means that on purely mathematical grounds, we can write 

But comparison with the fundamental thermodynamic relation, which contains the physics, we can make the following identifications: 

These (especially the second) are interesting in their own right. But we can go further, by differentiating both sides of the first equation by  and of the second by : 

Using the fact that the order of differentiation in the second derivation doesn't matter, we see that the right hand sides are equal, and thus so are the left hand sides, giving 

By starting with ,  and , we can get three more relations.

The two equations involving derivatives of  are particularly useful, as they provide a handle on  which isn't easily experimentally accessible.

For non-hydrodynamic systems, we can obtain analogous relations involving, say,  and  instead of  and ; for instance by starting with we get .