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**markwolfson16900** Hi, I'm working with different definitions of tangent vectors and trying to show they are equivalent. Here $\displaystyle M$ is a smooth manifold of dimension $\displaystyle n$ and $\displaystyle C^{\infty}$$\displaystyle (p)$ is the set of smooth, real valued functions defined on an open neighbourhood of $\displaystyle p \in M$.

Definition 1: A tangent vector at $\displaystyle p \in M$ is the tangent vector to some curve $\displaystyle c: (-\varepsilon ,\varepsilon ) \longrightarrow M$, i.e. a map $\displaystyle c'(0): C^{\infty}(p) \longrightarrow \mathbb{R}$ defined by $\displaystyle c'(0)(f) := \frac{\mathrm{d} (f\circ c)}{\mathrm{d} t} \bigg\vert_{t=0}$.

Definition 2: A tangent vector at $\displaystyle p \in M$ is a linear functional $\displaystyle X: C^{\infty}(p) \longrightarrow \mathbb{R}$ satisfying the Leibnitz product rule: $\displaystyle X(fg) = X(f)g(p) + f(p)X(g)$ for all $\displaystyle f,g \in C^{\infty}(p)$.

Definition 1 $\displaystyle \Longrightarrow$ Definition 2 is obvious. To show Definition 2 $\displaystyle \Longrightarrow$ Definition 1, let $\displaystyle (U,\phi )$ be a chart around $\displaystyle p$ with coordinate functions $\displaystyle x^1,\cdots , x^n$. Define $\displaystyle X^i = X(x^i)$ so that $\displaystyle X = X^i\frac{\partial}{\partial x^i} \Big\vert_p$.

Now define a curve $\displaystyle \gamma (t):= \phi^{-1} ( \phi(p) + t(X^1,\cdots ,X^n))$ so that, for any $\displaystyle f\in C^{\infty}(p)$, $\displaystyle \gamma '(0) (f) = \frac{\mathrm{d} f (\gamma (t))}{\mathrm{d} t} \Big\vert_{t=0} = X^i \frac{\mathrm{d} f\circ \phi^{-1}}{\mathrm{d} x^i} \Big\vert_{\phi (p)} = X^i \frac{\partial}{\partial x^i}\Big\vert_p (f)$. Hence $\displaystyle \gamma$ is the corresponding curve to $\displaystyle X$.

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 $\displaystyle X: C^{\infty}(p) \longrightarrow \mathbb{R}$ not satisfying the product rule)?

Any help would be appreciated.