Functional

A machine that eats an entire function and returns a single number -- arc length, energy, or travel time, for example.

A functional $J[y]=\int_a^b L(x,y,y')\,dx$ takes an entire function $y$ as input and returns a single number. The calculus of variations asks which function $y$ extremizes a given functional, exactly the way ordinary calculus asks which number extremizes a given function.

Examples

The arc-length functional $J[y]=\int_a^b\sqrt{1+(y')^2}\,dx$ returns a curve's length.