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 .