help with smooth manifolds

[franznietzsche]

i am trying to solve this problem:

Give the paraboloid y_{3}=(y_{1})^2+(y_{2})^2 the structure of a smooth manifold.

But i am unsure what it means by structure. Can anyone give me some help here?

[franznietzsche]

i kept trying to solve it on my own, but all i was able to get was:


x^1(y_{1},y_{2},y_{3})=\pm\sqrt{y_{3}-(y_{2})^2}



x^2(y_{1},y_{2},y_{3})=\pm\sqrt{y_{3}-(y_{1})^2}



x^3(y_{1},y_{2},y_{3})=(y_{1})^2 + (y_{2})^2


is that the answer? i'm feeling very lost here.

edit:

Ok i think i figured it out...

i set the local coordinates x^1,x^2 to lie in the y^1;y^3,y^2;y^3 planes respectively. thus from the above I get:


x^1(y_{1},0,y_{3})=\pm\sqrt{y_{3}}



x^2(0,y_{2},y_{3})=\pm\sqrt{y_{3}}


I also realised that the manifold is 2-dimensional (being embedded in E_{3} ) so there is no x^3 coordinate.


If any one can give any confirmation on this answer i would greatly appreciate it.