For a thermodynamic system there exists a function called entropy S(U,N,V) etc. We then define for instance temperature as: 1/T = ∂S/∂U μ = ∂S/∂N etc. When taking these partial it is understood that we only take the derivative of S wrt the explicit dependece on U,N etc. right? Because couldn't U carry an N dependence? I mean it does not for me make physical sense that the energy of the system should not be related to the number of particles in it. Actually it seems also a bit weird that there should be an explicit U dependence. Does this come from the fact that we are given the mean value of the internal energy of the system?