Definition of Functional in Calculus of variation

Question

I'm beginner in Calculus of variation. I do not understand why a derivative is compulsory in the definition of the functional, as we see $y'(x)$ in $$ J(y) := \int_a^b F(x,y(x),y'(x)) dx. $$ How does $y'(x)$ make $J(y)$ a functional?

Answer 1

A functional is simply a function that maps to $\Bbb{R}$. In this case the fact that this is a functional comes from the fact that you have an integral, returning a real number. The derivative comes from the nature of the Euler-Lagrange equation in your integrand and is not compulsory in the definition of a functional.