Hi, I'm working with different definitions of tangent vectors and trying to show they are equivalent. Here is a smooth manifold of dimension and is the set of smooth, real valued functions defined on an open neighbourhood of .
Definition 1: A tangent vector at is the tangent vector to some curve , i.e. a map defined by .
Definition 2: A tangent vector at is a linear functional satisfying the Leibnitz product rule: for all .
Definition 1 Definition 2 is obvious. To show Definition 2 Definition 1, let be a chart around with coordinate functions . Define so that .
Now define a curve so that, for any , . Hence is the corresponding curve to .
I have two questions. First of all: is this correct? Secondly: if so, then why have we not used the Leibnitz product rule (and so could this not be true for linear functionals not satisfying the product rule)?
Any help would be appreciated.
Let be a chart around with coordinate functions . Define so that . Now define so that, for any , . Hence .
My question then is: where do we use the fact that elements of satisfy the product rule? If we haven't used this fact, have we not defined an isomorphism between and the space of all linear functionals acting on (including ones which do not satisfy the product rule)?