Let be an open subset of a smooth manifold. Let be the collection of smooth functions . Let be the set of derivations , i.e. is linear and and . We define to be this space.

Now I am trying to show that if is open, with , then the linear map , given by for all is an isomorphism.

However I am not sure how to show it is injective. If we take , then , i.e. for all . But why does this tell us that for all ? Is every function in of the form for some ?

Any help with this would be appreciated.