Please help, I am just learning about manifolds, I need to know how to solve this problem:
Let M be the image of the application $\phi : \mathbb{R}^n \rightarrow \mathbb{R}^{2n} $
$\phi(u_1,u_2,...,u_n) = \frac{1}{1+ \displaystyle \sum_{i=1}^{n}u_i^2}(u_1,u_2,...,u_n,u_1^2,u_2^2,...,u_n^2) $
Show that M is a differentiable manifold and compute its dimension.