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.