Total differential of internal energy $U$ in terms of $p$ and $T$ using first law of thermodynamics in Fermi's Thermodynamics

Question

While reading pages 19-20 of Enrico Fermi's classic introductory text on Thermodynamics, I ran into two sources of confusion with his application of the First Law. Fermi introduces a peculiar notation which he exemplifies with the expression

$$ \bigg ( \frac{\partial U}{\partial T } \bigg )_{V}, $$

which he says,

"means the derivative of $U$ with respect to $T$, keeping $V$ constant, when $T$ and $V$ are taken as the independent variables" of a system "whose state can be defined in terms of any two of the three variables $V$, $p$, and $T$."

At first, I found this notation to be oddly redundant, since the usual definition of the partial derivative involves taking a "derivative with respect to one of those variables, with the others held constant." However, I'm aware that even in such simple cases as, e.g., a container of ideal gas with state equation $PV = Nk_{\rm{B}} T$, the state variables characterizing thermodynamic systems are often not independent.

1. Does this interdependence of the variables which determine the state of a thermodynamic system explain Fermi's introduction of this parenthetical notation for the partial derivative? If so, what nuance does this interdependence introduce that makes such notation necessary? If not, then why does Fermi use it?

Next, Fermi restates the First Law in differential form $\mathrm{d} U + \delta W = \mathrm{d} U + p~ \mathrm{d}V = \delta Q$ (written by Fermi as $dU + dL = dU + p~ dV = dQ$), then seems to use (something like) the definition of the total differential to say, "If we choose $V$ and $T$ as our independent variables...,"

$$ \mathrm{d} U = \bigg ( \frac{\partial U}{\partial T } \bigg )_{V} \mathrm{d} T + \bigg ( \frac{\partial U}{\partial V } \bigg )_{T} \mathrm{d} V,$$

which again seems fine to me, since he appears to be taking the total differential of $U = U(T,V)$ with respect to $T$ and $V$. Without actually working through the intermediate steps of their respective derivations, Fermi then states three identities (22), (23), and (24) for $\delta Q$ which vary according to which two variables among $T$, $p$, and $V$ are "chosen as the independent variables." Of the three equalities, (22) and (24) seem to be derived as

$$ \delta Q = \mathrm{d} U + p~\mathrm{d}V =\bigg ( \frac{\partial U}{\partial T } \bigg )_{V} \mathrm{d} T + \bigg ( \frac{\partial U}{\partial V } \bigg )_{T} \mathrm{d} V + p ~ \mathrm{d}V = \bigg ( \frac{\partial U}{\partial T } \bigg )_{V} \mathrm{d} T + \bigg [\bigg ( \frac{\partial U}{\partial V } \bigg )_{T} + p \bigg ] ~\mathrm{d} V$$

and

$$ \delta Q = \mathrm{d} U + p~\mathrm{d}V =\bigg ( \frac{\partial U}{\partial p } \bigg )_{V} \mathrm{d} p + \bigg ( \frac{\partial U}{\partial V } \bigg )_{p} \mathrm{d} V + p ~ \mathrm{d}V = \bigg ( \frac{\partial U}{\partial p } \bigg )_{V} \mathrm{d} p + \bigg [\bigg ( \frac{\partial U}{\partial V } \bigg )_{p} + p \bigg ] ~\mathrm{d} V,$$

which seem straightforward enough provided $( \partial X/ \partial Y)_Z$ really just denotes the usual partial derivative $\partial X / \partial Y$.

2. On the other hand, I have no idea how he arrived at (23),

$$ \delta Q = \bigg [\bigg ( \frac{\partial U}{\partial T } \bigg )_{p} + p \bigg ( \frac{\partial V}{\partial T } \bigg )_{p}~ \bigg ] ~\mathrm{d} T + \bigg [\bigg ( \frac{\partial U}{\partial p } \bigg )_{T} + p \bigg ( \frac{\partial V}{\partial p } \bigg )_{T}~ \bigg ] ~\mathrm{d} p,$$

which he claims is the differential form of the First Law when we take "$T$ and $p$ as the independent variables." How did Fermi obtain this identity?

Addendum following question closure: I'm not sure why this question was closed for lack of focus, as the two questions I asked were directly related to one another. I guess I failed to make my central concern sufficiently clear, namely that my failure to understand Fermi's notation (question 1) might have propagated forward into a failure to understand how he derived an identity (question 2 re (23)) using said notation. I think it would be nice if this question could be reopened in case anyone else wanted to contribute, but since I've already accepted Giorgio's answer, the stakes are pretty low here.

Answer 1

Fermi was probably the last physicist who excelled as a theoretician and an experimentalist. His lectures inspired many students, who became outstanding physicists in turn. His textbook on Thermodynamics is considered an example of his teaching style. It tries to condense the conceptual core of classical thermodynamics into a limited number of pages.

However, some of Fermi's approach's strengths can also be seen as flaws. In particular, Fermi is more interested in physical concepts than mathematically clean notation. In continuity with the previous tradition of the Thermodynamics textbook, he maintains a notation for thermodynamic quantities that would look ugly to a mathematician.

In particular, from the mathematical point of view, if we have a function of two variables, $U(V,T)$, its differential could be written without ambiguity as $$ dU = \frac{\partial U}{\partial T} dT + \frac{\partial U}{\partial V} dV, $$ without any need to stress with a subscript what variable the function is not derived on.

However, an equation of state connects pressure, volume, and temperature. This implies that the state represented by $V,T$ could be equally represented as $p$, and $V$, or $p$, and $T$, where $p=p(V,T)$.

From a mathematical point of view, we can use the equation of state to change the independent variables on which the internal energy depends. For example, if we use the equation of state to get the temperature as a function of volume and pressure, ($T=\tau(p,V)$) we can consider the compound function $$ \tilde U(p,V) = U(\tau(p,V),V).\tag{1} $$

I have used a different symbol for such a function of $p$ and $V$ because such a compound function is a different function of its independent variables than the original $U$. However, this is not the usual way such a change of variable is introduced in Thermodynamics. Based on the fact that the value of $\tilde U$ at $p,V$ is the same as the value of $U$ at the corresponding $V,T$ point,it is customary in Thermodynamics to keep the symbol $U$, even if the function is a different function.

At this stage, the notation with explicit information about what should be considered the second variable becomes useful and not redundant.

Clarified such a preliminary point, it should not be difficult to derive equation $(23)$ from $dU+pdV$ by taking into account that, in that case, the independent variables are intended to be $p$ and $T$.