Functionals and Extrema

Typical Problem: Consider a definite integral that depends on an unknown function y(x), as well as its derivative y'(x) = ${\frac{{{\rm d} y}}{{{\rm d} x}}}$,

I(y) = $$\int_{a}^{b}$$ F(x, y, y') dx.

A typical problem in the calculus of variations involve finding a particular function y(x) to maximize or minimize the integral I(y) subject to boundary conditions y(a) = A and y(b) = B.

Definition 1   The integral I(y) is an example of a functional, which (more generally) is a mapping from a set of allowable functions to the reals.

We say that I(y) has an extremum when I(y) takes its maximum or minimum value.