Let $$f: \mathbb{R}^{2} \times \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \quad f(z, w)=z w$$ be the map induced by the multiplication of complex numbers. Check whether it is a $C^{\infty}$-map.
By definition, $f$ is itself smooth ($C^k$) if its coordinate presentations $\hat{f} \equiv \psi \circ f \circ \varphi^{-1}$ with respect to any pair of charts happen to be smooth.
Note that $\mathcal{A} \equiv\left\{\mathcal{O}_{\alpha}, \varphi_{\alpha}\right\}$ is an atlas. It seems that we need to construct two $C^{\infty}$-maps and verify $\hat{f}$ is $C^{\infty}$-maps.
Any suggestions on how to approach this problem?
Thanks in Advance!
The identification of $\Bbb{C}$ with $\Bbb{R}^2$ is usually done by $z=x+iy\mapsto (x,y)$. Implicitly, you have used that to define a map $f:\Bbb{R}^2\times \Bbb{R}^2\to \Bbb{R}^2$ which is the map $\mu:\Bbb{C}\times \Bbb{C}\to \Bbb{C}$ given by $\mu(z,w)=zw$ in coordinates. So, use the criterion you mentioned above to check to see that the component functions are smooth. That is, if $z=x_1+iy_1$ and $w=x_2+iy_2$ then $$zw=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1).$$
Now, using this calculation can you see why your map is smooth? What are the component functions in local coordinates?