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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.

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    $\begingroup$ The question doesn’t make sense. You have two arbitrary vector fields in two arbitrary coordinate charts. Did you mean to write $X_p$ in the second equation? The question then makes sense and the answer is no, as the answer below says. $\endgroup$
    – Ted Shifrin
    Commented Mar 9 at 6:14
  • $\begingroup$ @TedShifrin yes I did mean to wrtie $X_p$, thanks for remindering me $\endgroup$
    – PermQi
    Commented Mar 9 at 6:36

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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.

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    $\begingroup$ Even easier … Just consider a linear change of coordinates on $\Bbb R^n$ for any $n$. $\endgroup$
    – Ted Shifrin
    Commented Mar 9 at 7:33

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