Functional and composite function

[Jhenrique]

What is the difference between a functional and a composite function?

Also, look those implicit equations: ##F(x, y(x))=0##, ##F(t, \vec{r}(t))=0##, ##F(x, y(x), y'(x), y''(x))=0##, ##F(t, \vec{r}(t), \vec{r}'(t))=0##... Can be understood that ##F## is the functional?

 

[Matterwave]

A composite function is a function of a function. The outer function takes 1 value from the inner function and produces another value. A functional takes the whole function (all the values that it takes) and produces a value. In other words, in a composite function, the outer function cares only about the inner function's value at a point, producing a different number for each different point. A functional only cares about the inner function's value everywhere, producing 1 single number for all different points.

It might be easier to explain with an example. Given a composite function ##g(f(t))## in order to return a number, we need to specify the value of ##t##. In other words ##g(f(2))## is a number while ##g(f(1))## is (potentially) a different number. Given a functional ##F[f(t)]## we need to specify the whole function ##f(t)## to give a number. In other words ##F[f(t)]## is already 1 number.

As for your follow up question. Those are usually functionals, yes, but where ambiguity might exist, the text should be clear about whether the object is a functional or a function. After all seeing ##F(f(t))## can be ambiguous. Often, texts will use square brackets to denote functionals (like I used above).