Next, let's look at how first-order ordinary differential equations can be used to specify a tangent field on a manifold.

There's a slight wrinkle here, because the idea of the tangent space and the tangent vectors that comprise it is actually _independent_ of any particular coordinate representation. Whatever charts we use to give coordinates to the points on, say a torus, we can still picture the tangent plane at a point (tangent space), and directions within it, apart from the specific coordinate representation.

Yet the ODEs are expressed in terms of coordinates.

There's a slight wrinkle here, because the idea of the tangent space and the tangent vectors that comprise it is actually _independent_ of any particular coordinate representation. Whatever charts we use to give coordinates to the points on, say a torus, we can still picture the tangent plane at a point (tangent space), and directions within it, apart from the specific coordinate representation.

Yet the ODEs are expressed in terms of coordinates.