Problem. Show that the restriction of $$Y = x_2 \frac{\partial}{\partial x_1} - x_1 \frac{\partial}{\partial x_2} + \dots + x_{2n} \frac{\partial}{\partial x_{2n-1}} - x_{2n-1} \frac{\partial}{\partial x_{2n}}$$ on $\mathbb{R}^{2n}$ to $S^{2n-1}$ defines a non-vanishing smooth vector field on $S^{2n-1}$.
My Attempt. I am confident in my proof that $Y|_{S^{2n-1}}$ is well-defined and non-vanishing. I'd like to ask whether or not my attempt to show it is smooth is correct:
To show that $Y|_{S^{2n-1}}$ is smooth, we write $\hat{Y}|_{S^{2n-1}}$, the coordinate representation of $Y|_{S^{2n-1}}$ and show that it is smooth in each component. So, if $p \in S^{2n-1}$ then there is some chart $p \in (U,\phi)$ where $\phi$ is the map defined, such that $\phi(x) = (x_1, \dots, \widehat{x_i}, \dots, x_{2n})$ and $\phi^{-1}(u) = (u_1, \dots, u_{i-1}, \sqrt{1 - |u|^2}, u_i, \dots u_{2n-1})$. Then $$\hat{Y}|_{S^{2n-1}}(u) = (u_1, \dots, \sqrt{1 - |u|^2}, \dots, u_{2n-1}, u_2, -u_1, \dots, \sqrt{1 - |u|^2}, -u_{i-1}, \dots, u_{2n-1}, - u_{2n-2})$$ if $i$ is even or $$\hat{Y}|_{S^{2n-1}}(u) = (u_1, \dots, \sqrt{1 - |u|^2}, \dots, u_{2n-1}, u_2, -u_1, \dots, u_{i+1}, - \sqrt{1 - |u|^2}, \dots, u_{2n-1}, - u_{2n-2})$$ if $i$ is odd. Either way, $\hat{Y}|_{S^{2n-1}}$ is smooth in each of its components, and so $Y|_{S^{2n-1}}$ is smooth.
Thanks so much!