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Math Help - Derivations

  1. #1
    Senior Member slevvio's Avatar
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    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.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Derivations

    Quote Originally Posted by slevvio View Post
    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?
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  3. #3
    Senior Member slevvio's Avatar
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    Re: Derivations

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