On Different Aspects of Differentiability and the Calculus of Variations

Abstract

We explore the notion of differentiability in finite and infinite dimensional spaces, including a geometric interpretation. This includes, in finite dimensions, the relationship between the existence of directional derivatives and that of a tangent hyperplane, and in infinite dimensions, Gˆateaux and Fr´echet differentiability. We apply these notions to the Calculus of Variations, and to finding the shortest path between two points on a flat plane and on a torus. To account for the domain, often a space of distributions, we define weak derivatives.