Ising model with a boundary magnetic field - an example of a boundary flow

In a nonperturbative proof of the g-theorem of Affleck and Ludwig was put forward. In this paper we illustrate how the proof of works on the example of the 2D Ising model at criticality perturbed by a boundary magnetic field. For this model we present explicit computations of all the quantities entering the proof including various contact terms. A free massless boson with a boundary mass term is considered as a warm-up example. (author)