text: Another way of deriving the Gibbs-Duhem equation can be found by taking the extensivity of energy into account. Extensivity implies that
  where  denotes all extensive variables of the internal energy . The internal energy is thus a first-order homogenous function. Applying Euler's homogeneous function theorem, one finds the following relation when taking only volume, number of particles, and entropy as extensive variables:
  Taking the total differential, one finds
  Finally, one can equate this expression to the definition of  to find the Gibbs-Duhem equation