- I know you can define a directional derivative on some subset of $\mathbb R^n$, but what can be said about an arbitrary set of points, $S$? What are the most general criteria $S$ must satisfy in order to define a directional derivative at each point in $S$? I suspect $S$ must define a differentiable manifold, but I quickly get lost in the terminology when I read the Wikipedia article on the subject.
- What about a generalized notion of a directional derivative? Say a set of points, $S$, must satisfy a set of criteria, $C$, in order to define a directional derivative, $D$, at each point in $S$. Is there any generalized notion of a directional derivative, $D'$, that applies to a set, $S'$, that satisfies a set of criteria, $C'\subset C$, such that $D'$ reduces to $D$ when the remaining criteria, $C-C'$, are applied to $S'$? In other words, is there a generalized directional derivative that the usual directional derivative is a special case of? If so, what criteria must apply to a set of points for there to be a well defined generalized directional derivative?
My understanding is that in many branches of math, there are many equivalent ways of defining things. If there is any chance these questions can be answered with relations between points and neighboring points, I would prefer that.
My math background:
My background in math (aside from "engineering math") is one course that used "baby Spivak" as the text book, and that was two years ago. Some of the intuition stuck, but the most of the rigor and definitions I learned have since faded.