I just want to make sure I understand differentials in the context of thermal physics. One of the big statements of thermodynamics is the conservation of energy in terms of the state variables [itex]U,T,S,V,P[/itex]: [tex] dU = T dS - P dV. [/tex] What does this really MEAN though? Is there any way to understand this in a mathematically rigorous way? People keep explaining it to me like this: "Well, what is means is that a SMALL change in energy is equal to a SMALL change in entropy times..." But "small" is a relative term...so what does "small" mean in this context?