# Is the component function of a smooth vector field unique?

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.

• 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. Commented Mar 9 at 6:14
• @TedShifrin yes I did mean to wrtie $X_p$, thanks for remindering me Commented Mar 9 at 6:36

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