ts maps preserve its linear and differentiable structure.
| Part of the problem of constructing an atlas is that a map must be inverted
| and that its composition with another map be a diffeomorphism.  Reducing our
| example from a sphere to a circle simplifies this calculation considerably.
|
| Next, preparatory to considering the general construction of a tangent space,
| the notion of equivalence classes of curves on a manifold, and their addition
| and scalar multiplication is explored.  This study prov