Equivalence of Definitions of Tangent Vector

Theorem
Let $M$ be a smooth manifold.

Let $m \in M$ be a point.

Let $V$ be an open neighborhood of $m$.

Let $\map {C^\infty} {V, \R}$ be defined as the set of all smooth mappings $f: V \to \R$.

Definition 2 implies Definition 1
Let $\lambda \in \R$ and $f, g \in \map {C^\infty} {V, \R}$.

Thus $X_m$ is linear.

Hence $X_m$ satisfies the Leibniz law.

Thus $X_m$ satisfies Definition 1.

Lemma 1
Let $X_m$ be a tangent vector at $m \in M$ according to Definition 1.

Let $V$ be an open neighborhood of $M$.

Let $f \in \map {C^\infty} {V, \R}$ be constant.

Then $\map {X_m} f = 0$.

Proof of Lemma 1
Let $\map f m = 0$.

Then, by constancy, $f = 0$ on $V$.

Hence, by linearity, $\map {X_m} 0 = 0$.

Let $\map f m \ne 0$.

$f$ is constant, $\exists \lambda \in \R : f \sqbrk V = \set \lambda$  $f = \lambda$.

Let $X_m$ be a tangent vector at $m \in M$ according to Definition 1.

Denote $n := \dim M$.

Let $f \in \map {C^\infty} {V, \R}$.

Let $\struct {U, \kappa}$ be a chart such that $\map \kappa m = 0$.

Let $\kappa^i$ be the $i$th coordinate function of the chart $\struct {U, \kappa}$.

By, Taylor's Theorem/n Variables :

Observe that $\ds \map {\paren {f \circ \kappa^{-1} } } 0 = \map {\paren {f \circ \kappa^{-1} } } {\map \kappa m} = \map f m$ is a constant mapping on $V$.

Define $X^i := \map {X_m} {\kappa^i}$.

Then by linearity:
 * $\ds \map {X_m} f = \map {X_m} {\map f m} + \sum_{i \mathop = 1}^n \map {\frac {\map \partial {f \circ \kappa^{-1} } } {\partial \kappa^i} } 0 \ X^i + \map {X_m} {\map {\OO_2} \kappa}$

Lemma 2

 * $\map {X_m} {\map {\OO_2} \kappa} = 0$

Proof of Lemma 2
By Taylor's Theorem/n Variables, for each summand of $\map {\OO_2} \kappa$ there exists $i \in \set {1, \ldots, n}$ and an $h \in \map {C^\infty} {V, \R}$ with $\map h m = 0$ such that the summand is $\kappa^i h$.

Thus the sum $\map {\OO_2} \kappa$ vanishes.

By Lemma 1:
 * $\map {X_m} {\map f m} = 0$

By Lemma 2:
 * $\map {X_m} {\map {\OO_2} \kappa} = 0$

Hence:
 * $\ds \map {X_m} f = \sum_{i \mathop = 1}^n \map {\frac {\map \partial {f \circ \kappa^{-1} } } {\partial \kappa^i} } 0 \ X^i$

Let $\set {e_i}$ be a basis of $\R^n$ such that:
 * $\ds \kappa = \sum_{i \mathop = 1}^n \kappa^i e_i$

Choose a smooth curve $\gamma: I \to M$ with $0 \in I \subseteq \R$ such that $\map \gamma 0 = m$ and:


 * $\map {\dfrac {\d \kappa^i \circ \gamma} {\d \tau} } 0 := X^i$

Then:

Hence $X_m$ is a tangent vector according to Definition 2.

This proves the assertion.