Now it is clear to see how ODEs can define a tangent vector field on a manifold.

For this, we use first-order, explicit ODEs in \\(\mathbb{R^n}\\), which give the tangent vector in \\(\mathbb{R^n}\\) as a function of the coordinate point \\(C(s) \in \mathbb{R^n}\\).

Then just send those tangent vectors in \\(\mathbb{R^n}\\) through the aforementioned vector-space isomorphisms, i.e. through the charts, to get the tangent vectors for the manifold proper.

For this, we use first-order, explicit ODEs in \\(\mathbb{R^n}\\), which give the tangent vector in \\(\mathbb{R^n}\\) as a function of the coordinate point \\(C(s) \in \mathbb{R^n}\\).

Then just send those tangent vectors in \\(\mathbb{R^n}\\) through the aforementioned vector-space isomorphisms, i.e. through the charts, to get the tangent vectors for the manifold proper.