The extra structure that we can extract from the chart is a correspondence between the tangent space in the manifold and the associated tangent space in the parameter space \\(\mathbb{R^n}\\).

It is a vector-space isomorphism between \\(\mathit{Tangent}(s)\\) and \\(\mathit{Tangent}(C(s)) = \mathbb{R^n}\\).

It is a vector-space isomorphism between \\(\mathit{Tangent}(s)\\) and \\(\mathit{Tangent}(C(s)) = \mathbb{R^n}\\).