Let $M$ be a smooth manifold. We know that if $X:\ M\to TM$ is a smooth vector field, then for any local chart $x=(x_1,\dots,x_n):\ U\subset M\to\mathbb R^n$, there exists smooth functions $X_1,\dots,X_n:\ U\to\mathbb R $ such that
$$X_p\,=\,\sum X_i(p)\frac{\partial }{\partial x_i}\bigg|_p\ \ \forall p\in U $$
I wonder if $y:\ V\subset M\to\mathbb R^n $ is another local chart such that $U\cap V\ne\varnothing $, and
$$X_p\,=\,\sum Y_i(p)\frac{\partial }{\partial y_i}\bigg|_p\ \ \forall p\in V $$
then is it true that $X_i(p)=Y_i(p)\ \ \forall p\in U\cap V$ ?
May you give me some hints for this question ? Thanks.
Are you asking if two coordinate charts would always give the same components for a vector at a point? If so, I think considering cartesian and polar coordinates on $\mathbb{R}^2$ should answer your question. The vector field itself is manifestly invariant but the component functions which describe the vectors at the various points will obviously change in general.