I have problems in writing down a vector field $X$ on the sphere $S^2$ such that the integral curves of $X$ are the meridians of $S^2$.
I proceed in this way.
I consider this parametrisation of $S^2$: $$ x=\cos(\varphi) \cos(\alpha),$$$$ y=\cos(\varphi) \sin(\alpha), $$ $$ z=\sin(\varphi) .$$
A meridian is a curve: $$\sigma(t)=(\sigma_1(t),\sigma_2(t),\sigma_3(t)) =(\cos(\varphi+t) \cos(\alpha),\cos(\varphi+t) \sin(\alpha),\sin(\varphi+t)).$$
Then I have that: $$\dot\sigma_1(t)=\cos(\alpha)\sin(\varphi+t)$$ $$\dot\sigma_2(t)=-\sin(\alpha)\sin(\varphi+t)$$ $$\dot\sigma_1(t)=\cos(\varphi+t)$$
Now, in order to find the components of the vector field, should I calculate these derivates for $t=0$ ? In this way is the vector field written in respect to the basis $\partial x$, $\partial y$, $\partial z$?
My ideas are quite confused. Thanks for the help!