# Manipulating Formulas with Derivatives

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

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Hurkyl
Staff Emeritus
Gold Member
One way to make sense of what they're doing is via differential forms. If the sequence xi 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 xi'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)

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