Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^n,\ \forall\ p \in M \}.$$
Choosing an atlas $\{(\varphi_i,U_i)\}_{i=1}^{n}$, and compacts $K_i \subset U_i$, such that $$\bigcup_{i=1}^n K_i = M,$$ we define the $\|\cdot \|_r$ norm as
\begin{align*}\|\cdot\|_r : \mathfrak{X}(M)&\to \mathbb{R}\\ X &\to \max_{\substack{i\in\{1,...,n\} \\ j\in \{0,...,r\}}}\left\{\sup_{x \in \varphi^{-1}_i(K_i)}\left\| \text{d}^{j}\left( X\circ\varphi_i \right) \right\|\right\}, \end{align*}
then we named $\mathfrak{X}^r(M)$ as the complete Banach space $(\mathfrak{X}(M),\|\cdot\|_r)$ (it is possible to prove that the topology of $\mathfrak{X}^r(M)$ does not depend on the selected atlas).
My Question: Let $X \in \mathfrak{X}(M)$ and $Y$ be a smooth vector field on $M$ defined just in a compact $K \subset M$ such that $$\max_{\substack{i\in\{1,...,n\} \\ j\in \{0,...,r\}}}\left\{\sup_{x \in \varphi^{-1}_i(K_i\cap K)}\left\| \text{d}^{j}\left( X\circ\varphi_i \right) - \text{d}^{j}\left( Y\circ\varphi_i \right) \right\|\right\}<\varepsilon,$$ is it possible extend $Y$ to a vector field $\tilde{Y}$ such that
1) $\left.\tilde{Y}\right|_{K} = Y$,
2) $\|X-\tilde Y\|_r < A\cdot\varepsilon$ , where $A $ is a constant that depends only on the manifold $K$ ?
The compact $K$ is a connected submanifold with boundary of $M$, such that $\dim K = \dim M$.
Edit: I changed $\|X-\tilde Y\|_r < \varepsilon$ to $\|X-\tilde Y\|_r < A\cdot\varepsilon$ after Moishe Kohan's comment.
My ideas
First, I extend $Y$ by a smooth vector field $Z$ $\in \mathfrak{X}(M)$, by the continuity of $Z$, so there exists a neighborhood $U$ of $K$, such that $$\max_{\substack{i\in\{1,...,n\} \\ j\in \{0,...,r\}}}\left\{\sup_{x \in \varphi^{-1}_i(K_i\cap U)}\left\| \text{d}^{j}\left( X\circ\varphi_i \right) - \text{d}^{j}\left( Z\circ\varphi_i \right) \right\|\right\}<\varepsilon,$$
and then choosing a partition of unity $ \{\phi_1, \phi_2\}$ subordinate to the cover $\{U,M\setminus K\}$ we can define $$\tilde{Y} = \phi_1 Z + \phi_2 X, $$ however I could not guarantee that $\|X - \tilde{Y}\|_r < \varepsilon$, because I can not control de derivatives of $\phi_1$ and $\phi_2$. Does anyone know how should I proceed?
