Local representation of jet of manifold

Let $M$ be $C^{\infty}$-manifold and $(U,\phi)$ be local chart with $x\in U$ where $\phi =(x_1, \dots,x_n)$. For $X\in (T_k^rM)_x:=J_0^r(\mathbb{R}^k,M)_x$, can the jet $X$ be represented by the local chart like a tangent vector? And if it can be, how do we represent it?

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 Yes. The local chart is just the set of the jets from a neighbhorhood of $0 \in \mathbb{R}^k$ to $U$. I suggest you work out the details for $k = 1$ first. – Deane Yang Sep 8 at 19:29