A definition of derivation at point of \(C^k\) functions
Topics
About: \(C^\infty\) manifold with boundary
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of germ of \(C^k\) functions at point.
Target Context
- The reader will have a definition of derivation at point of \(C^k\) functions.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any \(C^\infty\) manifold with (possibly empty) boundary, \(M\), and any point, \(p \in M\), any \(\mathbb{R}\)-linear map, \(D_v: C^k_p (M) \to \mathbb{R}\), that satisfies the Leibniz rule, \(D_v (f_1 f_2) = D_v (f_1) f_2 (p) + f_1 (p) D_v (f_2)\)
2: Note
Strictly speaking, \(C^\infty\) manifold with boundary is not required for \(k \neq \infty\), but just topological manifold with boundary does not suffice, because \(C^k\)-ness is not defined there.
