Lecture 3 - Part 1 of 3 - Total (Frechet) Derivative of Functions and Functionals (First Variation)
This lecture uses the previously developed theory of Normed Linear Spaces to introduce the concept of the Total Derivative (Frechet Derivative) of a functional. I start by reviewing and expanding on the concept of the total derivative of a single variable function and a multivariable function. Once this is complete, I extend the definition to the Total Derivative of a functional. Finally, I use the Total Derivative to derive the first variation of an integral functional that is typical in the Simplest Variational problem. This generates a result which is more general (but still equivalent) to the result obtained previously using Eulers Method of finite differences.