How do we compute the differential of a smooth map restricted on an embedded submanifold?

Question

As an example, consider the set $Z:=\{(x,p,q)\in\mathbb{R}^3|x^3+px+q=0\}$. This is an embedded submanifold since its defining map $\Phi(x,p,q)=x^3+px+q$ has $0$ as a regular value. Now define projection map $\pi:Z\rightarrow \mathbb{R}^2$ by $(x,p,q)\mapsto (p,q)$. This is the restirction of the projection map $\pi:\mathbb{R}^3\rightarrow \mathbb{R}^2$ on $Z$. I would like to compute $d\pi$ restricted on the tangent spaces of $Z$. I have tried to compute the differential of the composition $\pi\circ\iota$, where $\iota:Z\rightarrow\mathbb{R}^3$ is the inclusion map. But this does not work at all since I could not write down the explicit formula for $\iota$. Any hint will be appreciated.

Answer 1

Since $Z = \Phi^{-1}(0)$ and $0$ is a regular value of $\Phi$, a standard theorem says that $T_{(x,p,q)}Z = \ker d_{(x,p,q)}\Phi = \ker \begin{bmatrix}3x^2+p & x & 1\end{bmatrix} = \left\lbrace \begin{bmatrix}u \\ v \\ -(3x^2+1)u-xv \end{bmatrix}\mid (u,v)\in\mathbb{R}^2 \right\rbrace$. Hence $$ d_{(p,q)}\pi\vert_{T_{(x,p,q)}Z}:\begin{bmatrix}u \\ v \\ -(3x^2+1)u-xv \end{bmatrix} \mapsto \begin{bmatrix}v \\ -(3x^2+1)u-xv \end{bmatrix}. $$