So I have had related questions over the past month, but I would like to ask this question to clarify my understanding. In thermodynamics you work with certain potentials, which are a function of the thermodynamic variables, i.e.: U(S,T,V,N,P) Now for U one has the identity: dU= TdS+SdT-pdV+VdP etc etc. From these one figure out relations like: T = dU/dS at fixed V,T,P... It is this thing about the thermodynamic variables being fixed that has always confused me. In general are the thermodynamic variables S,P,T,V,N not correlated? How am I to understand then the derivative if I am keeping the other thermodynamic variables fixed. Consider for instance including particles of different kinds: U = ... + μ1N1 + μ2N2 Now we have that: μ1 = dU/dN1 at fixed T,S,V, N2 But how can I keep N2 fixed if I am in a resevoir, where adding a particle to one phase with N1 particles, actually takes away a particle from the other i.e. dN1=-dN2 Similarly, if I change for instance V, don't I change S or P etc etc.