which is the Dirichlet energy of ϕ ϕ . Classical extrema of the action (the solutions to the Lagrange equations) are then those field configurations that minimize the Dirichlet energy of ϕ ϕ . Another way to convert this expression into a more easily-recognizable form is to observe that, for a scalar function f : M → → R one has d ∗ ∗ f = 0 and so one may also write