Is this a valid parametrization of the torus?

[hideelo]

I am reading "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo and on page 156 he gives the following parameterization of the torus

x(u,v) = ((a + r cos u )cos v, (a + r cos u)sin v, r sin u) 0 < u < 2*pi, 0 < v < 2*pi

Doesnt this leave out some of the torus,? I know that he needs to map from an open set in R^2 and he therefore want to map from the product of two open intervals in R, but by not including either 0 or 2*pi I don't see how this is onto the torus. Am I missing something here?

 

[Ben Niehoff]

Yes, strictly speaking it covers all of the torus except two intersecting circles.

 

[hideelo]

But then it's not covering it, why is this not a problem?

 

[Ben Niehoff]

You can't cover the torus in a single chart.