# Calculus of Variations

Calculus of variation is concerned with maximizing or minimizing certain types of functions given in the form of integrals called functionals.

We need to find a function $y(x)$ such that the integral

$I$ = $\int_{x_{1}}^{x_{2}}$ $f(x, y, y')dx$

will be a maximum or minimum.

Let $S$ be a set of functions of a variable $x$ defined over an interval $(a, b)$. Then a function which assigns to every element in $S$ a unique real number is called a functional. In the 18th century Euler and Lagrange laid the foundations to calculus of variations with the classical problems of determining a closed curve in the plane.

Domain of a functional is a set of admissible functions. In ordinary functions the values of the independent variables are numbers. Whereas with functionals, the values of the independent variables are functions.