Show that
(1) M= $ \{(x,y,z) \in \mathbb{R}^{3}$ :$ x^{2}+y^{2}+z^{2} - 3xyz = 1 \}$ is a submanifold of $\mathbb{R}^{3}$.
(2) Compute the Tangent Space of M at $a$, for $a \in M$.
Attempt(1) If I define $f$ such that $f(x,y,z) = x^{2} + y^{2}+z^{2} - 3xyz $
Then if $\Delta(f) \ne 0$ then M is a submanifold of $\mathbb{R}^{3}$
If I let it $= 0$ i get,
$x = (3/2) yz$
$y= (3/2)xz$
$z= (3/2)xy$
(2) I am stuck here and finding it hard to get notes to help (or notes I can understand).
$T_{a}M$ is the set of all derivations on all of the equivalence classes in $C^{\infty} (M,a)$. Do I have to create another map that agrees on $M$?
Any help here would be appreciated.
The gradient is $${\rm grad}\ f(x,y,z)=[2x-3yz,2y-3xz,2z-3xy],$$ which is $[0,0,0]$ iff $x=0,y=0,z=0$.
Since this critical point is not in the level surface then the surface is regular.
Let us take a point in the level surface $f^{-1}(1)$, let say $a=(1,1,\frac{3+\sqrt 5}{2})$.
We get ${\rm grad}\ f(a)=[-\frac{5+\sqrt 5}{2},-\frac{5+\sqrt 5}{2},\frac{-3+\sqrt 5}{2}]$.
Then the equation for a plane, tangent to the level surface at $a$, is $$-\frac{5+\sqrt 5}{2}x-\frac{5+\sqrt 5}{2}y+\frac{-3+\sqrt 5}{2}z=C$$ for some constant $C$.