I'm working on a problem that asks me to describe a differentiable manifold structure for the circle $S^1$ and then to calculate the restriction of the vector field $$\xi=-y\frac{\partial}{\partial x }+x\frac{\partial }{\partial y}$$ to $S^1$ in local coordinates (given by my description of the differentiable manifold structure)
My atlas for the circle is given by the charts $(U_1,\phi_1),(U_2,\phi_1),(U_3,\phi_1),(U_4,\phi_1)$ where $U_1,U_2,U_3,U_4$ are the upper, lower, left and right open semi-circles, respectively and $\phi_1(x,y)=x,\phi_2(x,y)=x,\phi_3(x,y)=y,\phi_4(x,y)=x$. I've already check that this charts are pairwise compatible so this is a correct description of the differentiable structure of $S^1$. However, I don't understand how to calculate the restriction of the vector field. I think in example 1.1 of this document something similar is done but with polar coordinates but I don't understand very well how. Any help would be appreciated