I am trying to understand the definition of a lie derivative of a smooth function with respect to a vector field. Here are the relevant definitions.
If $M,N$ are smooth manifolds and $f:M\to N$ is a smooth map, then $d_pf : T_pM \to T_{f(p)}N$ , $[γ] \mapsto [f\circ γ]$ is the differential of $f$ at $p$. Here $T_pM$ is the tangent space of M at p.
Let $M$ be a smooth manifold, $TM$ the tangent bundle of $M$, and $\pi : TM\to M$ the canonical projection, i.e. $\pi$ assigns to each tangent vector $X\in T_pM$ its foot point $p\in M$. A vector field on $M$ is a smooth map $X:M\to TM$ with $π\circ X=id_M$.
If $f:M\to \mathbb{R}$ is a smooth function, the Lie derivative of $f$ with respect to a vector field $X$ on $M$ is the function $\mathcal{L}_Xf :M\to \mathbb{R}$, $p\mapsto d_pf(X_p)$.
Where I get confused is what $X_p$ is supposed to mean in this definition. I have carefully scanned my notes and there is nothing about what the vector field with a subscript $p$ could mean. Any ideas what it could possibly mean?