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.