5.1 Functionals and stationary points

As will be illustrated below, we can generalise the concept of a function to that of a functional, a mapping of a function onto a number. An example could be

where we have also introduced a “square-brackets notation” to distinguish this from a function. In short, a functional I[y] has a definite numerical value for each function x → y(x).

Example 5.1: 

An example of a functional is

I[y] ={\mathop{ \mathop{\mathop{∫ }\nolimits }}\nolimits }_{0}^{π}y(x)\kern 1.66702pt dx.

Some illustrative values of the functional are

\array{ y(x) &I[y] ̲ ̲ \cr \mathop{sin}\nolimits x&2\cr \mathop{cos}\nolimits x&0 \cr x &{π}^{2}∕2 \cr {x}^{2} &{π}^{3}∕3\cr \mathop{\mathop{⋮}}&\mathop{\mathop{⋮}} }