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 U in R^n there are no constraints, so the tangent space at a point p is another copy of R^n. The set U could be a coordinate chart for an n-dimensional manifold.

The tangent space at p, denoted TM_p, is the set of possible velocity vectors of paths through p. Hence there is a canonical vector basis: if (x_1,...,x_n) are the coordinates, then v_1,...,v_n are a basis for the tangent space, where v_i is the velocity vector of a particle with unit speed moving inward along the coordinate x_i. 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 M.

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

Tangent space stretch

For example, let U=(0,1) and V=(0,3) be coordinate charts for the unit interval I. We can change coordinates with phi:U->V defined by phi(x)=x+2x^2. This is a change of coordinates because the derivative does not vanish on U. But this change is not linear, and stretches out I more near 1 than it does near 0. The tangent vectors transform by the derivative. At x=1/4, they are stretched by a factor of dphi/dx=2. While at x=3/4, they are stretched out by a factor of dphi/dx=4.

In general, the tangent vectors transform according to the Jacobian. The tangent vector v at q can also be considered as the tangent vector J_phiv at phi(q) in another coordinate chart, where phi is the diffeomorphism from one chart to the other. The linear transformation determined by the Jacobian of phi is invertible, since phi 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 f:X->Y between manifolds, it makes sense to map the tangent vectors of X to tangent vectors of Y. Writing f^~ as the function f between a coordinate chart in X and one in Y, then f_*(v)=J_(f^~)(v) maps v from TX_p to TY_(f(p)). Another notation for f_* is df, the differential of f. In the language of tensors, the tangent vector's pushing forward means that a vector field is a covariant tensor.

See also

Calculus, Coordinate Chart, Differential k-Form, Directional Derivative, Euclidean Space, Exterior Algebra, Jacobian, Manifold, Submanifold, Tangent Bundle, Tangent Space, Vector Field, Velocity Vector

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Chart Tangent Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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