The second law has been shown to be equivalent to the internal energy U defined as a convex function of the other extensive properties of the system. That is, when a system is described by stating its internal energy U, an extensive variable, as a function of its entropy S, volume V, and mol number N, i.e. U = U (S, V, N), then the temperature is equal to the partial derivative of the internal energy with respect to the entropy (essentially equivalent to the first TdS equation for V and N held constant):