Chart Tangent Space

From the point of view of coordinate charts, the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In an open set  in  there are no constraints, so the tangent space at a point  is another copy of . The set  could be a coordinate chart for an -dimensional manifold.

The tangent space at , denoted , is the set of possible velocity vectors of paths through . Hence there is a canonical vector basis: if  are the coordinates, then  are a basis for the tangent space, where  is the velocity vector of a particle with unit speed moving inward along the coordinate . The collection of all tangent vectors to every point on the manifold, called the tangent bundle, is the phase space of a single particle moving in the manifold .

It seems as if the tangent space at  is the same as the tangent space at all other points in the chart . However, while they do share the same dimension and are isomorphic, in a change of coordinates, they lose their canonical isomorphism.

For example, let  and  be coordinate charts for the unit interval . We can change coordinates with  defined by . This is a change of coordinates because the derivative does not vanish on . But this change is not linear, and stretches out  more near 1 than it does near 0. The tangent vectors transform by the derivative. At , they are stretched by a factor of . While at , they are stretched out by a factor of .

In general, the tangent vectors transform according to the Jacobian. The tangent vector  at  can also be considered as the tangent vector  at  in another coordinate chart, where  is the diffeomorphism from one chart to the other. The linear transformation determined by the Jacobian of  is invertible, since  is a diffeomorphism.

Not only does the Jacobian, and the chain rule, show that the tangent space is well-defined, independent of coordinate chart, but it also shows that tangent vectors "push forward." That is, given any smooth map  between manifolds, it makes sense to map the tangent vectors of  to tangent vectors of . Writing  as the function  between a coordinate chart in  and one in , then  maps  from  to . Another notation for  is , the differential of . In the language of tensors, the tangent vector's pushing forward means that a vector field is a covariant tensor.