1. Derivations

Let $U \subseteq M$ be an open subset of a smooth manifold. Let $C^{\infty}(U)$ be the collection of smooth functions $f: C^{\infty}(U) \rightarrow \mathbb{R}$. Let $\operatorname{Der}_p (C^{\infty}(U))$be the set of derivations $v: C^{\infty}(U) \rightarrow \mathbb{R}$, i.e. $v$ is linear and and $v(fg) = f(p)v(g) + g(p)v(f)$. We define $T_p M$ to be this space.

Now I am trying to show that if $W \subseteq U$ is open, with $p \in W$, then the linear map $r: \operatorname{Der}_p(C^{\infty}(W)) \rightarrow \operatorname{Der}_p (C^{\infty}(U))$, given by $r(v)(f) = v (f|_W)$ for all $f \in C^{\infty}(U)$ is an isomorphism.

However I am not sure how to show it is injective. If we take $v \in \operatorname{ker}r$, then $r(v) = 0$, i.e. $r(v)(f) = v(f|_W) = 0$ for all $f\in C^{\infty}(U)$. But why does this tell us that $v(f) = 0$ for all $f \in C^{\infty}(W)$? Is every function in $C^{\infty}(W)$ of the form $f|_W$ for some $f \in C^{\infty}(U)$ ?

Any help with this would be appreciated.

2. Re: Derivations

Originally Posted by slevvio
Let $U \subseteq M$ be an open subset of a smooth manifold. Let $C^{\infty}(U)$ be the collection of smooth functions $f: C^{\infty}(U) \rightarrow \mathbb{R}$. Let $\operatorname{Der}_p (C^{\infty}(U))$be the set of derivations $v: C^{\infty}(U) \rightarrow \mathbb{R}$, i.e. $v$ is linear and and $v(fg) = f(p)v(g) + g(p)v(f)$. We define $T_p M$ to be this space.

Now I am trying to show that if $W \subseteq U$ is open, with $p \in W$, then the linear map $r: \operatorname{Der}_p(C^{\infty}(W)) \rightarrow \operatorname{Der}_p (C^{\infty}(U))$, given by $r(v)(f) = v (f|_W)$ for all $f \in C^{\infty}(U)$ is an isomorphism.

However I am not sure how to show it is injective. If we take $v \in \operatorname{ker}r$, then $r(v) = 0$, i.e. $r(v)(f) = v(f|_W) = 0$ for all $f\in C^{\infty}(U)$. But why does this tell us that $v(f) = 0$ for all $f \in C^{\infty}(W)$? Is every function in $C^{\infty}(W)$ of the form $f|_W$ for some $f \in C^{\infty}(U)$ ?

Any help with this would be appreciated.
Let me give you a hint, and see if you can figure it out from there. What is the other descrption of $T_pM$ and how does this relate to the last post I helped you with?

3. Re: Derivations

Can't this be proved purely algebraically, using the fact that the function f is smooth?