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I have a homework problem in Introductory Differential Geometry, i feel somewhat confident I understand what I have to do, but I don't know how to do the necessary steps:

Given a map $F: \mathbb{S^3} \rightarrow \mathbb{S^2}$ such that $F(u, w) = (u\bar{w} + w\bar{u}, iu\bar{w} - iw\bar{u}, w\bar{w} - u\bar{u})$,

View $\mathbb{S^3}$ as $\{ (u,w ) \in \mathbb{C^2} : |u|^2 + |w|^2 = 1 \}$, compute sufficiently many coordinate representations of $F$ to show that it is smooth.

The coordinate representation that came to my mind was using stereographic coordinates to view $\mathbb{S^3}$ with two charts, one without the north pole and one without the south pole and the maps $\sigma_{1,2}: \mathbb{S^3}\big/(N,S) \rightarrow \mathbb{R^3}$.

Then to check that $F$ is smooth, according to the theory we covered in class, I only need to check that $F\circ \sigma_{1,2}^{-1}$ ?

I think I am getting confused when thinking of $\mathbb{S^3}$ as a subset of $\mathbb{C^2}$, can I instead think $\mathbb{C^2} \simeq \mathbb{R^4}$ and try to use my approach with stereographic coordinates, or is there something better to do?

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    To simplify your life somewhat, you might contemplate this. If you have a smooth function $G\colon\Bbb C^2 \to \Bbb R^3$ and $G$ maps points of $S^3$ to $S^2$, can you deduce that $G\big|_{S^3} = F$ is smooth as a map $S^3\to S^2$? – 2017-02-03

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If you are asked to show that $F$ is smooth by using coordinate representations (also known as parametrizations) then what you suggest will work. Note only that both the domain $\mathbb{S}^3$ and the codomain $\mathbb{S}^2$ of $F$ are submanifolds so you need to use parametrizations $\sigma_i \colon U_i \rightarrow \mathbb{R}^3$ for the domain and parametrizations $\psi_i \colon V_i \rightarrow \mathbb{R}^2$ for the codomain and check that $\psi_j \circ F \circ \sigma_i^{-1} \colon \mathbb{R}^3 \rightarrow \mathbb{R}^2$ are smooth for $1 \leq i,j \leq 2$.