Calculus of variations

Summary

The calculus of variations studies the extreme values of functionals, that is, of maps that send functions to numbers. Examples of functionals are length, area and energy. A typical functional is an integral $\int_\Omega L(x, u, \nabla u) \, dx$, minimised among functions $u$ with given boundary values. Its smooth critical points solve a partial differential equation, the Euler–Lagrange equation. Conversely, many equations of physics and geometry are Euler–Lagrange equations, and the calculus of variations gives existence through the direct method.