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In the context of
dE = T ds - P dV [1a]
I use in that eq dU as internal energy not dE or energy
gravitational energy is outside dU
To me that seems like one step forward and two steps back. For
one thing, U is not conserved, so to do anything useful you need
to constantly switch back and forth between U and E.
Secondly,
the definition of E is reasonably well known, whereas it’s hard
to learn and hard to remember what’s supposed to be included in
U. In situations where the height h is relevant, you will have
to keep track of it one way or another, so the only question is
whether you will need one new variable or two (h alone versus
h and U both).