Four years after taking a thermo course and I finally understand what the hell the partial derivative notation in thermodynamics means I think
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The notation for partial derivatives is actually ambiguous. Yes, it implies that all the other variables are held constant, but what ARE the other variables? Consider f(x,y)=(x+y)²; what is ∂f/∂x? It depends, are you holding y constant or z=x+y constant?
why would you hold x+y constant? once you define f(x,y) then ∂f/∂x is unambiguous
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But this is exactly what we DON'T do in thermodynamics! Do you distinguish between S(E,V), S(E,P), S(T,V)...? It's just S.
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When i took thermo we *always* wrote S(E,V), S(E,P), S(T,V) etc..
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The idea is that the function f' defined implicitly by f(x,y(x,z)):=f'(x,z) has a different value for its x derivative.
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Speaking from a position of probable ignorance, is it just me or is the standard notation for partial derivatives just the worst? It requires that you remember what role the variables played when you defined the function with no accepted notation for making it explicit, meaning..
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