A Brief Introduction to Calculus of Variations

To find a stationary point (i.e., a local maximum or minimum) the derivative ḟ(x) needs to vanish. In higher dimensions, the gradient ∇f needs to be identical to zero. The latter may also be expressed as fu = 0 ∀u. In a sense these simple derivatives are the most basic form of the calculus of variations. Usually, however, one talks about calculus of variations in the context of determining functions that minimize functionals. In other words, the argument (for which a given quantity, the functional) is extremal is no longer a point in space, but a function out of an appropriate class of functions. Given a functional J : F 7→ R (where F is the desired space of functions) the Gâteaux variation is (compare this to the directional derivative)