Let \( M \) be a smooth manifold and \( p \in M \). A tangent vector at \( p \in M \) is an equivalence class of smooth curves
\[ \{ c : (-\varepsilon, \varepsilon) \to M \mid c(0) = p \}, \]where \( c_1 \sim c_2 \) if there exists a smooth coordinate map \( \varphi \) such that the coordinate representations of \( c_1 \) and \( c_2 \) with respect to \( \varphi \),
\[ \bar{c}_1 = \varphi^{-1} \circ c_1, \quad \bar{c}_2 = \varphi^{-1} \circ c_2, \]satisfy
\[ \bar{c}_1(0) = \bar{c}_2(0) \quad \text{and} \quad \bar{c}_1'(0) = \bar{c}_2'(0). \]