> [!definition] > > Let $X$ be a Hausdorff [[Manifold|manifold]] of class $C^p$, $TX$ with $\pi: TX \to X$ be its [[Tangent Bundle|tangent bundle]], a **vector field** on $X$ is a [[Sections of Vector Bundles|section of]] the tangent bundle/a [[Manifold Morphism|morphism]] of class $C^{p - 1}$ > $ > \xi: X \to TX > $ > such that $\xi(p) \in T_pX$ is in its [[Tangent Space|tangent space]] for all $p \in X$. The set $\vf(X)$ is the space of all vector fields on $X$. > > If $\xi$ is a right-inverse of $\pi$, but is not a morphism, then $\xi$ is a *rough* vector field. # Coordinate Representations > [!definition] > > Let $X$ be a [[n-Manifold|n-manifold]], let $\xi: X \to TX$ be a rough vector field, $p \in X$, and $(U, (x^i))$ be a coordinate [[Atlas|chart]] at $p$, then $\xi$ has coordinate representation > $ > \xi(p) = \xi^i(p)\ppip > $ > with $\xi^i: U \to \real$ being a **component functions of** $\xi$ in the given chart.