Calculus of variation is a branch of mathematics that deals with the study of functions, functionals, and their respective extremal problems. It specifically focuses on finding the functions that minimize or maximize functionals, which are expressions or mappings that assign numerical values to functions.
The subject explores the properties and behavior of these extreme functions, as well as the techniques required to find them. It offers a mathematical framework to address optimization problems in various fields, including physics, engineering, and economics.
At its core, the calculus of variation is concerned with defining and computing variations of a functional – that is, the change in its value based on infinitesimal perturbations applied to the function it acts on. It aims to discover the specific conditions under which a functional will have an extremum or extremal function.
The calculus of variation utilizes concepts from calculus and differential equations, alongside the theory of Hilbert spaces and functional analysis. It employs techniques such as Euler-Lagrange equations, which establish necessary conditions for optimality, and the method of variations, which involves manipulating functionals to obtain extremal solutions.
In summary, the calculus of variation is a mathematical discipline encompassing the study of functions and their optimization through the analysis of functionals. It provides tools and principles to determine the extremal behavior of these functions and plays a vital role in diverse scientific and engineering applications.
