Determining the differential of a map defined on a submanifold of $\Bbb R^n$

Question

Let $M=\{(x,y,z,w)\in \Bbb R^4:x^3+y^3+z^3-3xyz=1\}$ and consider the function $f:M\to \Bbb R^2$ defined by $f(x,y,z,w)=(x+y+z+w,w^3+w)$. It is easily checked that $1$ is a regular value of the function $g:\Bbb R^4\to \Bbb R$, $(x,y,z,w)\mapsto x^3+y^3+z^3-3xyz$, so $M$ is a regular submanifold of $\Bbb R^4$, of dimension $3$. I am trying to determine the differential $f_*:T_p(M)\to T_{F(p)}\Bbb R^2$ at a point $p=(x_0,y_0,z_0,w_0)\in M$. I have shown that tangent space $T_pM \subset T_p\Bbb R^4=\Bbb R^4$ is precisely the set $\{v\in \Bbb R^4: \text{grad}g(p)\cdot v=0\}$. But then how do I have to compute the differential of $f$ at $p$?

Answer 1

Hints:

$1).\ $ Fix $(x_0,y_0,z_0,w_0)\in M$, and find a slice chart $(\varphi, U)$ for $M$ about $(x_0,y_0,z_0,w_0)$. Try letting $F(x,y,z,w)=x^3+y^3+z^3-3xyz-1$ and taking $\varphi=(F,y,z,w).$

$2).\ $Now check that

$$\begin{pmatrix} F_x &F_y &F_z &F_w \\ 0&1 &0 &0 \\ 0& 0&1 &0 \\ 0&0 &0 &1 \end{pmatrix}=\begin{pmatrix} 3x^2-3yz&3y^2-3xz &3z^2-3xy &0 \\ 0&1 &0 &0 \\ 0& 0&1 &0 \\ 0&0 &0 &1 \end{pmatrix}$$

is nonsingular at $(x_0,y_0,z_0,w_0).$

$3).\ $ Find the matrix representation of $f_*$ by using the standard coordinates $(\psi, V)$ on $\mathbb R^2$ about $f(x_0,y_0,z_0,w_0)$, and computing the Jacobian matrix of $\psi\circ f\circ \varphi^{-1}$ at $\varphi(x_0,y_0,z_0,w_0)$.