this is Loring W.Tu " An introduction to manifolds" Proposition 17.9
I understand the proof. But, Is it necessary to "By Proposition 14.4, we can extend the $C^\infty$ vector field $X$ on U to a $C^\infty$ vector field on M" ? Why we must extend $X$ on U to vector field on M ?
Many thanks!
Because what you're assuming is that $\omega(X)$ is smooth for any smooth vector field $X$ on $M$. It doesn't state anything about vector fields on $U$.
So it doesn't tell you anything about $\omega\left(\frac{\partial}{\partial x_j}\right)$, unless you can somehow extend $\frac{\partial}{\partial x_j}$ to a smooth vector field $X$ on $M$, so that you can apply your assumption. The proposition gives you sort of an extension: for any point $p\in U$, you can define $X\in\mathfrak{X}(M)$, which agrees with $\frac{\partial}{\partial x_j}$ in some open set $V_p^j$ rather than the whole set $U$, but that's still enough for what you want. Then your assumption gives you that $\omega(X)$ is smooth and so the restriction
$$\omega(X)\big|_{V_p^j}=\omega\left(\frac{\partial}{\partial x_j}\right)\Big|_{V_p^j}$$
is also smooth.