Submanifold of $\mathbb{R}^3$

Question

If $M:=\{(x,y,z)\in \mathbb{R}^3|f(x,y,z)=x^2+y^2+z^2+xy+yz+zx=1\}$, then $M$ is submanifold of $\mathbb{R}^3$.

I think it is difficult to make local coordinate system (I tried by solving $f$, but I couldn't). So is there another way to check it?

Answer 1

$p:=(x,y,z)$ and Jacobi matrix $J_f(p)=(2x+y+z,x+2y+z,x+y+2z)$ Then $rank J_f(p)=0$, $p=(0,0,0)$,so $p \not \in M$ . It means $1$ is regular value of $f$. And $M=f^{-1}(\{1\})$, by implicit function theorem, $M$ is submanifold.