f(x).  If now f : X -&gt; Y is a morphism of
| one manifold into another, and x a point of X, then by means of charts
| we can interpret the derivative of f on each chart at x as a mapping
|
| df(x)  =  T_x f : T_x(X) -&gt; T_f(x)(Y).
|
| Indeed, this map T_x f is the unique linear map having the following property.
| If (U, q) is a chart at x and (V, r) is a chart at f(x) such that f(U) c V
| and ^v^ is a tangent vector at x represented by v in the chart (U, q), then
|
| T_x f(^v^)
|
| is the tangent vecto