functional

An ordinary function takes in a number and gives back a number: feed it x, it returns f(x). A functional takes in a whole function and gives back a single number. Picture every possible road between two towns; assign to each road its length, or the time a ball takes to roll down it, or the energy stored in a wire bent into that shape. That assignment — one number per entire curve — is a functional. It is a machine whose input is not a point but a path.

Most functionals in this subject are integrals. A typical one is J[y] = integral from a to b of L(x, y, y') dx, where the integrand L (the Lagrangian, or running cost) depends on the position x, the height of the curve y(x), and its slope y'(x). To evaluate J you must already have the entire curve y on hand, slope and all; you then add up L all along it. The square brackets J[y] are the standard notation reminding you that the argument is a function, not a number. The arc length integral from a to b of sqrt(1 + y'^2) dx is the simplest example: it reports the length of the curve y.

Functionals are the central object of the calculus of variations, because the laws of physics and the best designs of engineering are almost always statements that some functional is as small (or as stationary) as possible: least time, least energy, least action, shortest path, minimal surface area. Asking which input function makes a functional smallest is a vastly bigger search than ordinary optimization — the unknown is an entire curve, an infinite-dimensional object, not a handful of numbers. That is the whole reason this field needs its own calculus.