Let $M$ be a smooth manifold and $\varphi=(x^1,...,x^n)$ local coordinates defined on $U$. Then the coordinate vector fields $$(\varphi^{-1})_*\frac{\partial}{\partial x^i}$$ determines smooth vector fields on $U$. Can these always be smoothly extended to $M$? (It may vanish outside $U$.) I think they should, in order to make other definitions like the connection and curvature tensor valid. I consider the books of John M. Lee.
I know that a general smooth vector field on an open set of $M$ cannot always be smoothly extended, take for example: $X:(0,1)\to T\mathbb R, x\to \frac 1{x(x-1)}\frac{\partial}{\partial x^1}$.