Now there's even more structure that can be extracted from the charts.

Choose a point \\(s \in M\\), and let \\(C(s) \in \mathbb{R^n}\\) be its coordinate representation.

In addition to the tangent space \\(\mathit{Tangent(s)}\\) at \\(s\\) in the manifold itself, there is also a tangent space \\(\mathit{Tangent(C(s))}\\) around \\(C(s) \in \mathbb{R^n}\\).

That's because \\(\mathbb{R^n}\\) is itself a simple kind of manifold, at so at every point in \\(\mathbb{R^n}\\) there is a vector space of tangent vectors at that point.

Now that tangent space \\(\mathit{Tangent}(C(s))\\) turns out to be just a copy of \\(\mathit{R^n}\\), but in principle we still visualize as a local copy of \\(\mathbb{R^n}\\) at the point \\(C(s))\\).

Choose a point \\(s \in M\\), and let \\(C(s) \in \mathbb{R^n}\\) be its coordinate representation.

In addition to the tangent space \\(\mathit{Tangent(s)}\\) at \\(s\\) in the manifold itself, there is also a tangent space \\(\mathit{Tangent(C(s))}\\) around \\(C(s) \in \mathbb{R^n}\\).

That's because \\(\mathbb{R^n}\\) is itself a simple kind of manifold, at so at every point in \\(\mathbb{R^n}\\) there is a vector space of tangent vectors at that point.

Now that tangent space \\(\mathit{Tangent}(C(s))\\) turns out to be just a copy of \\(\mathit{R^n}\\), but in principle we still visualize as a local copy of \\(\mathbb{R^n}\\) at the point \\(C(s))\\).