Consider map $g:SO_2\times \mathbb R^2\to \mathbb R^2$, $g(A,v)=Av$, where $A\in SO_2$ is an orthogonal $2\times 2$ matrix and $v\in \mathbb R^2$ is a $2$-vector.
Show that $0$ is a critical value.
My try;
If we paremetrize $SO_2$ by \begin{array}{|cc|} \cos(t) & \sin(t) \\ -\sin(t) & \cos(t) \\ \end{array} then we will get paremetrization of $SO_2\times \mathbb R^2$ by $r(t,u,v)=(\cos(t),\sin(t),-\sin(t),\cos(t),u,v)$ for $t,u,v\in \mathbb R$.
Hence, $\text{Rank}(J(g\circ r))=\text{Rank} J(g)$ where $J(f)$ denotes the Jacobian matrices of $f$. Thus, I found that $0$ is not a critical value which is wrong according to question.