I'm trying to solve the following question:
Given a smooth manifold $M^m$, a point $p \in M$, a point $x \in \mathbb{R}^n$ and a linear map $L:T_pM \to \mathbb{R^n}$. Prove that exists a smooth map $f:M^m \to \mathbb{R}^n$ such that $f(p)=x$ and $df(p)=L$.
Any idea, solution or reference for this problem
Let $U$ be an open subset of $M$ with $p \in U$ and $\varphi:U\to \mathbb R^n$ a diffeomorphism. Let $\chi:M\to \mathbb R$ be a smooth map with values in $[0,1]$, equal to $0$ outside of $U$ and to $1$ in a neighborhood of $1$.
$T_p\varphi$ is a isomorphism $T_pM\to \mathbb R^n$. Then, set : $$\forall x \in M, f(x) = \left\{\begin{array}{cl} \chi(x) L\cdot (T_p\varphi)^{-1}\cdot\varphi(x) & \text{if }x\in U\\ 0 & \text{if }x\notin U \end{array}\right.$$ We compute : $$T_pf = L\cdot (T_p\varphi)^{-1}\cdot T_p\varphi = L$$