Verify that a set is a submanifold I'm stuck on the following exercise:
"Verify that $M:=\{(x,y,z)\in\mathbb{R}^3:x^2+3y^2+2z^2=3, x+y+z=0\}$ is a submanifold. Also, say what its dimension is and compute the Jacobian matrix of the function."
The definition of $C^k (k\geq 1)$ submanifold of dimension $p$ that I have in my notes is:
"$V\subset\mathbb{R}^n$ is a $C^k (k\geq 1)$ submanifold of dimension $1\leq p\leq n-1$ iff $\forall x_0\in V \exists U$ neighbourhood of $x_0$ in $\mathbb{R}^n$ and a function $f\colon U\to\mathbb{R}^{n-p}$, $f\in C^k(U)$ such that: (1) $x_0\in U$, (2) $V\cap U=\{x\in U:f(x)=0\}$, (3)rank $Jf(x)=n-p\ \forall x\in U$."
It's the first exercise I do on (sub)manifolds and I don't know how to get started, so I would appreciate if someone explained to me how to do this kind of exercise.
Best regards,
lorenzo.

You have a set in $\mathbb{R}^3$ defined by 2 equations, so, as in linear algebra, you can start guessing that it will have dimension 1. If you are familiar with the equations, you will know that your are cutting a cone with a plane, which results in a curve.
Following your definition, I would define the function 
$f(x,y,z)= ( x^2+3y^2+2z^2-3, x+y+z )$.
It is obvious that $f(p)=0$ if and only if $p\in M$. Now you have to check the other conditions.
