So I'm new to differential geometry and this problem is giving me trouble, and more generally I'd just like to understand pushforwards of vector fields better. Let $\phi: \mathbb{S}^2\rightarrow\mathbb{R^2}$, $\phi(x,y,z) = (\frac{x}{1-z}, \frac{y}{1-z})$ and $\phi: \mathbb{S}^2\rightarrow\mathbb{R^2}$, $\psi(x,y,z) = (\frac{x}{1+z}, \frac{y}{1+z})$, and let $X$ be a vector field on $\mathbb{S^2}$ satisfying $\phi_*(X)(u,v) = (1,0)$ for all $(u,v) \in \mathbb{R}^2$. We want to find $\psi_*(X)(u,v)$.
I really don't know how to start here. I understand the definition of the pushforward of a vector field, but it just seems so abstract and hard to do actual calculations with. Can anyone offer some insight?