Manipulating Formulas with Derivatives

[Bernie Hunt]

The Problem;
Given H = U + PV and dU = TdS - PdV
Find dH in terms of T, S, P, V

My Solution;

H = U + PV
dH = dU + PdV + VdP
dH = (TdS - PdV) + PdV + VdP
dH = Tds + VdP

My Question

Am I missing a step between the first and second steps? I'm taking the derivative of both sides, but not specifying what the derivative is in respect to. (bad English, sorry)

I learn this short hand from some physics guys, but I'm looking for the strict mathematical method.
Any suggestions?

Thanks,
Bernie

 

[Hurkyl]

One way to make sense of what they're doing is via differential forms. If the sequence x_i are generalized coordinates on (a local patch of) the state space, so that any function f on the state space can be represented as a function of the x_i's, then one property of differential forms is that

$$df = \sum_i \frac{\partial f}{\partial x_i} dx_i,$$

which, formally, looks just like the chain rule. Because of the formal similarity, differentials share many of the same properties as derivatives, such as

d(fg) = f dg + g df.​

(Some formulations of differential forms take this property as part of the definition)



(In differential geometry, it is customary to write i as a superscript, not a subscript. But I wrote it this way becuase I figured it was probably more familiar to you. In particular, so that it doesn't look like exponentiation)

 

[Bernie Hunt]

Thanks for your reply Hurkyl.

I haven't had DE yet, so I can't really comment on your reply.

Bernie