Understanding smooth manifolds I'm having a little trouble understanding smooth manifolds.
I have been told the ellipsode $E = x^2+2y^{2}+3z^{2}=6$ Where $(x,y,z) \in\mathbb{R}^3$ is a 2-dimensional manifold of smoothness but I didn't really understand why. Can somebody help explain why?
Thanks

You don't have to know what a manifold is in order to check that a subset of $\mathbb R^n$ is a submanifold!
To see that $E$ is a submanifold it suffices to check that the smooth function $f(P)=f(x,y,z)= x^2+2y^{2}+3z^{2}-6$ has a non-zero gradient at all  $P\in E$, which is clear since $\operatorname {grad} f(x,y,z)=(2x,4y,6z)$ only vanishes at the origin. 
An important but neglected distinction
a) Certain subsets  $M\subset \mathbb R^n$ are called submanifolds of $\mathbb R^n$: there are several equivalent definitions for these sets, all deriving from the implicit function theorem.
The point I want to emphasize is that (in principle!) given  $M$ one can answer the question "is $M$ a submanifold of $\mathbb R^n$ ?" unequivocally with "yes!" or "no!", without invoking any extraneous structure.   
b) On the other hand, given an abstract set  $S$ it does not make sense to ask whether $S$ is or is not a smooth manifold: a smooth manifold is a structure consisting of of a topology on  $S$ plus a complicated set $\mathcal A$, called an atlas, which has to satisfy a lot of weird requirements. 
c) The link between the two concepts above is that a submanifold  $M\subset \mathbb R^n$ can be endowed very canonically with such an atlas $\mathcal A$, so that $(M,\mathcal A)$ becomes a manifold in its own right.
The important practical point however  is that  a student asked to check whether some $M\subset R^n$ is a manifold should never worry about atlases or charts: submanifolds are automatically provided with a canonical atlas and the student should only concentrate on the tools for proving that $M$ is a submanifold, like implicit function theorem, submersions, gradients, differentials,  immersions, diffeomorphisms,... (these depend of course on the presentation the teacher has chosen).   
