Скачать или смотреть Minimization In Infinite Dimensions With The Calculus Of Variations

I believe that the best way to understand minimization in infinite dimensions is to first carefully study minimization in finite dimensions. In finite dimensions, we can minimize functions using either partial derivatives or directional derivatives. If we want to minimize functionals, i.e., functions whose inputs are infinite dimensional, however, we cannot define a basis and thus we cannot define partial derivatives. This means that we must use directional derivatives to formulate the necessary condition for a minimum. I think that motivating the use of directional derivatives in this way is the best way to introduce the calculus of variations.

After introducing the basic concepts of the calculus of variations, I show how the calculus of variations is related to differential equations. Specifically, many differential equations like the Poisson equation can be interpreted as the necessary condition for a minimum of an energy functional. With the tools provided by the calculus of variations, the weak formulation of the Poisson equation can be identified as the necessary condition of the minimization problem.

Video about the weak formulation:    • I Finally Understood The Weak Formulation ...  

0:00 Introduction
2:32 Partial Derivatives and Directional Derivatives
9:38 Functionals
15:56 Minimizing Functionals
19:22 The Calculus of Variations and Differential Equations
22:50 Remarks on Notation
25:07 Summary

Corrigendum:
24:31 The alternative formula for the directional derivative is incorrect. It should be d/dh F[u + hv] |h=0.

Keywords: fundamental theorem of the calculus of variations, variational calculus, partial derivative, directional derivative, first variation, functional derivative, variational derivative, infinite-dimensional, functional, function norm, test function, virtual function, virtual field, finite element method, finite element analysis, partial differential equation, Poission problem, weak formulation, principle of virtual work, principle of virtual displacement, Euler-Lagrange equations

Music: Swans In Flight - Asher Fulero