where c o n s t is an arbitrary positive constant and where R is the universal gas constant. It is easily seen that U is a linearly homogeneous function of the three variables (that is, it is extensive in these variables), and that it is weakly convex. Knowing temperature and pressure to be the derivatives T = ∂ ∂ U ∂ ∂ S , P = − − ∂ ∂ U ∂ ∂ V , the ideal gas law P V = N R T immediately follows as below: