How to calculate the tangent space for some concrete examples? I don't understand how to calculate the tangent space of some concrete examples. For example, can someone explain me how to calculate the tangent space in a point $p$ on the sphere? And how to calculate the tangent space in a point on the torus? 
 A: In general if you have a manifold $M$ (already) embedded in $\mathbb{R}^N$, as $M = \mathbb{S}^n$,then  for a given point $p$, you can consider a chart $(U,\varphi:U\to \mathbb{R}^n)$, $p\in U$ and thus  $ T_p M 
 = d_{\varphi(p)}\varphi^{-1} (T_{\varphi(p)}(\mathbb{R}^n)) $.
In practice, you choose a local parametrization $\varphi^{-1}:\mathbb{R}^n\to M$ and computing partial derivatives $d_{x}\varphi^{-1}(e_i)\in T_{\varphi^{-1}(x)}M$ will give you $n$ generators of the tangent space.
Since the manifold is given as a subset of $\mathbb{R}^N$, $d_{x}\varphi^{-1}(e_i)$ will be also vectors of $\mathbb{R}^N$, so they are something concrete. If you consider an abstract manifold, you will obtain again some generators but is not so simple to visualize the tangent space.
Another useful remark is the following. If you are dealing with a level set  in the form $M = \{ x\in \mathbb{R}^N | \  f(x ) = O \} \ $ where $f:\mathbb{R}^N\to \mathbb{R}^k $  and the point $O$ is regular for $f$ (this is again the case of $\mathbb{S}^n$ or $\mathbb{T}^2$), then it can be proven that $T_pM = \ker d_pf $.
So since $\mathbb{S^n} = \{x\in \mathbb{R}^{n+1}| \  ||x||^2 - 1 = 0\}$ and 
$0$ is regular for $||x||^2 - 1$,  it follows that
$$ T_p \mathbb{S}^n= \ker d_p(||\cdot||^2 - 1) = \ker 2\langle p,\cdot\rangle = p^\perp$$
