From the expression $$U=TS - PV + \mu N$$ $U$ can be described as a function of three of these six variables: $$U=U(S,V,N),$$ implying that the other variables are also functions of these; i.e: $$T=T(S,V,N), P=P(S,V,N), \mu = \mu(S,V,N)$$ In particular, $$T=\partial U/ \partial S, P= \partial U/ \partial V, \mu = \partial U/ \partial N.$$

Now one would naively expect the follow differential relation to hold: $$dU = TdS + SdT - PdV -VdP +\mu dN + N d\mu.$$ However, the thermodynamic identity is written: $$dU = TdS - PdV -VdP +\mu dN.$$ What is the meaning of this? It seems to necessitate: $$SdT-VdP+ Nd\mu =0$$ which may seem clear when viewed from the point of view of extensive vs. intensive variables? Expanding the above, $$Sd(\partial U/ \partial S)-Vd(\partial U/ \partial V)+ Nd(\partial U/ \partial N) =0$$ $$S(\frac{\partial^2 U}{\partial S \partial S}dS+ \frac{\partial^2 U}{\partial S \partial V }dV + \frac{\partial^2 U}{\partial S \partial N}dN) - V(...)+ N (...)=0$$ Which is not very illuminating.

From the expression $$U=TS - PV + \mu N$$ $U$ can be described as a function of three of these six variables: $$U=U(S,V,N),$$ implying that the other variables are also functions of these; i.e: $$T=T(S,V,N), P=P(S,V,N), \mu = \mu(S,V,N)$$ In particular, $$T=\partial U/ \partial S, P= \partial U/ \partial V, \mu = \partial U/ \partial N.$$

Now one would naively expect the follow differential relation to hold: $$dU = TdS + SdT - PdV -VdP +\mu dN + N d\mu.$$ However, the thermodynamic identity is written: $$dU = TdS - PdV +\mu dN.$$ What is the meaning of this? It seems to necessitate: $$SdT-VdP+ Nd\mu =0$$ which may seem clear when viewed from the point of view of extensive vs. intensive variables? Expanding the above, $$Sd(\partial U/ \partial S)-Vd(\partial U/ \partial V)+ Nd(\partial U/ \partial N) =0$$ $$S(\frac{\partial^2 U}{\partial S \partial S}dS+ \frac{\partial^2 U}{\partial S \partial V }dV + \frac{\partial^2 U}{\partial S \partial N}dN) - V(...)+ N (...)=0$$ Which is not very illuminating.