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.
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