TABLE OF CONTENTS From the analytic point of view, geometric quantities such as length, area, volume, are regarded as the value determined by the function parametrizing the object, and therefore, each of them induces a mapping from the set of functions into R. Sometimes, such a mapping is called the functional because it is a function defined on function spaces. In the calculus of variation, a functional is given, and it is required to find its extremal functions. This formulation can describe physical problems if the functional is taken as energy, Lagrangian, free energy, and so forth.
The Jordan curve indicates a closed non-self-intersecting curve, and a connected open set is referred to as the domain. We can observe that a Jordan curve ? on the plane R 2 encloses there a simply connected domain D. The question studied here is referred to as the isoperimetric problem. When is the area A of D minimized if the length L of ? is prescribed?
The answer is a circle. Analytic proof is as follows. First, we parametrize ? as (x(t), y(t)) in t ? [ a, b]. This implies that ( x(a), y(a)) = ( x(b), y(b)),
and
Putting a 1( x, y) = y, a 2( x, y) = 0 in the Green's formula (1.43), we obtain
Here, we take the parametrization t =...
