Let $n=(0,0,1)$ and $s=(0,0,-1)$ and let $\phi:\mathbb S^2\setminus\{n\}\to\mathbb{R}^2$ and $\psi:\mathbb S^2\setminus\{s\}\to\mathbb{R}^2$ be the stereographic projections given by $(u,v)=\phi(x,y,z)=\big({x\over 1-z}, {y\over 1-z}\big)$ and $(r,s)=\psi(x,y,z)=\big({x\over 1+z}, {y\over 1+z}\big)$. Suppose that $X$ is a smooth vector field on $\mathbb S^2$ with $(\phi_*X)(u,v)=(1,0)$ for all $(u,v)\in\mathbb{R}^2$.
Find the vector field $(\psi_*X)(r,s)$.
For each $(c,d)\in\mathbb{R}^2$, find the integral curve of $\psi_*X$ at $(c,d)$, that is find the smooth map $\beta:\mathbb R\to\mathbb R^2$ such that $\beta(0)=(c,d)$ and $\beta'(t)=(\psi_*X)(\beta(t))$ for all $t\in\mathbb R$.
For part (1), I calculated the vector field of the pushforward to be \begin{align*} \psi_* X(r, s) & = \left( \frac{s^2-r^2}{\left(r^2+s^2\right)^2},-\frac{2 r s}{\left(r^2+s^2\right)^2}\right) \end{align*} I am not sure if my calculation is correct for part (1). But assuming it does, then for part (2). It seems like that I would need to solve an partial differential equation?
I have no prior experience on solving PDEs and I have no clue on what to do.
Any hints are appreciated!
The definition I am using is:
Let $M\subseteq \mathbb{R}^n$ be a smooth regular submanifold, and let $X$ be a smooth vector field on $M$. An integral curve of $X$ on $M$ at $p$ is a smooth map $\gamma:J\subseteq\mathbb{R}\to M$ with $\gamma(0)=p$ (where $J$ is an open interval with $0\in J$) such that $$\gamma'(t)= X(\gamma(t))\ \hbox{ for all }\ t\in J.$$ Let $\sigma:U_\sigma\subseteq\mathbb{R}^m\to V_\sigma\subseteq M$ be a chart at $p$ with $\sigma(a)=p$. Let $\alpha:I\subseteq\mathbb{R}\to U_\sigma\subseteq\mathbb{R}^m$ be a smooth map with $\alpha(0)=a$, and let $\gamma(t)=\sigma(\alpha(t))$. Let $A_\sigma$ be the smooth vector field on $U_\sigma$ such that $X(\sigma(u))= D\sigma(u) A_\sigma(u)$ for all $u\in U_\sigma$.
Note that $\gamma'(t)=D\sigma(\alpha(t))\alpha'(t)$ and that $X(\gamma(t))=X(\sigma(\alpha(t)))= D\sigma(\alpha(t))\, A_\sigma(\alpha(t))$.
I suppose my question is similar to this question: Why tangent vectors rotate 720 degrees after stereographic projection from the sphere?
I tried to graph the vector field of the pushforward. And it looks like this?
Your pushforward looks plausible up to scalar multiplication (both algebraically, and geometrically from your plot), but on algebraic grounds should be a polynomial vector field, and on geometric grounds should have a zero at the origin rather than a pole of order two. (The constant vector field $(1, 0)$ fixes the point at infinity in the Riemann sphere.)
Finding flow lines generally entails only solving an ODE, but here it's a coupled, non-linear system. The sneaky trick is, you can write down the flow of the constant vector field $(1, 0)$ by inspection; composing with $\psi \circ \phi^{-1}$ gives the flow of the pushforward vector field "because chain rule".