A definition of differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of differential of \(C^\infty\) map between \(C^\infty\) manifolds with boundary at point.
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 \(C^\infty\) manifolds with boundary, \(M_1, M_2\), any \(C^\infty\) map, \(f: M_1 \to M_2\), and any point, \(p \in M_1\), the map, \(d f_p: T_pM_1 \to T_{f (p)}M_2\), \(v \mapsto d f_p (v)\), such that for any \(f' \in C^\infty (M_2)\), \(d f_p (v) (f') = v (f' \circ f)\)
2: Note
\(d f_p (v)\) is indeed a derivation, because for any \(r \in \mathbb{R}\), \(d f_p (v) (r f') = v (r f' \circ f) = r v (f' \circ f) = r d f_p (v) (f')\), being \(\mathbb{R}\) linear; \(d f_p (v) (f'_1 f'_2) = v ((f'_1 f'_2) \circ f) = v ((f'_1 \circ f) (f'_2 \circ f)) = v (f'_1 \circ f) (f'_2 \circ f (p)) + (f'_1 \circ f (p)) v (f'_2 \circ f) = d f_p (v) (f'_1) f'_2 (f (p)) + f'_1 (f (p)) d f_p (v) (f'_2)\), satisfying the Leibniz rule.
