All the tangent spaces of a manifold may be ‘glued together’ to form a new differentiable manifold with twice the dimension of the original manifold, called the "tangent bundle" of the manifold. The informal description above relies on a manifold's ability to be embedded into an ambient vector space formula_11 so that the tangent vectors can ‘stick out’ of the manifold into the ambient space. However, it is more convenient to define the notion of a tangent space based solely on the manifold itself. There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below. In the embedded-manifold picture, a tangent vector at a point formula_1 is thought of as the "velocity" of a curve passing through the point formula_1. We can therefore define a tangent vector as an equivalence class of curves passing through formula_1 while being tangent to each other at formula_1. Suppose that formula_16 is a formula_17 manifold (formula_18) and that formula_19. Pick a coordinate chart formula_20, where formula_21 is an open subset of formula_16 containing formula_1. Suppose further that two curves formula_24 with formula_25 are given such that both formula_26 are differentiable in the ordinary sense (we call these "differentiable curves initialized at formula_1"). Then formula_28 and formula_29 are said to be "equivalent" at formula_30 if and only if the derivatives of formula_31 and formula_32 at formula_30 coincide. This defines an equivalence relation on the set of all differentiable curves initialized at formula_1, and equivalence classes of such curves are known as "tangent vectors" of formula_16 at formula_1. The equivalence class of any such curve formula_37 is denoted by formula_38. The "tangent space" of formula_16 at formula_1, denoted by formula_41, is then defined as the set of all tangent vectors at formula_1; it does not depend on the choice of coordinate chart formula_20. To define vector-space operations on formula_41, we use a chart formula_20 and define a map formula_46 by formula_47 where formula_48. Again, one needs to check that this construction does not depend on the particular chart formula_20 and the curve formula_37 being used, and in fact it does not.