One can attach to every point of a differentiable manifold a tangent space, a real vector space that intuitively contains the possible “directions” at which one can tangentially pass through . The elements of the tangent space are called tangent vectors at. All tangent spaces of a connected manifold have the same dimension, equal to the dimension of the manifold
Definition as velocities of curves
Suppose is a manifold () and is a point in . Pick a chart where is an open subset of containing . Suppose two curves and with are given such that and are both differentiable at . Then and are called equivalent at if the ordinary derivatives of and coincide. This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vector of at . The equivalence class of the curve is written as . The tangent space of at , denoted by , is defined as the set of all tangent vectors, it does not depened on the choice of chart .
To define the vector space operations on , we use a chart and define the map . It turns out that this map is bijective and can thus be used to transfer the vector space operation from over , turning the latter into an -dimensional real vector space. Again, one need to check that this construction does not depend on the particular char chosen, and in fact is does not.
Definition via derivations
Suppose that is a manifold. A real valued function belongs to if is infinitely differentaible for every chart . is a real associative algebra for the pointwise product and sum of functions and scalar multiplication.
Pick a point in . A derivation at is a liner map that has the property that for all in :
modeled on the product rule of calculus. If we define addition and scalar multiplication for such derivations by
we get a real vector space which we define as the tangent space .
The relation between the tangent vectors defined earlier and derivatoin is as follows: if is a curve with tangent vector , then the corresponding derivation .