a vector field serves to define a generalized ordinary differential equation on a manifold: A solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. 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. Formal definitions The informal description above relies on a manifold's ability to be embedded into an ambient vector space 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. Definition via tangent curves In the embedded-manifold picture, a tangent vector at a point is thought of as the velocity of a curve passing through the point . We can therefore define a tangent vector as an equivalence class of curves passing through while being tangent to each other at . Suppose that is a differentiable manifold (with smoothness ) and that . Pick a coordinate chart , where is an open subset of containing . Suppose further that two curves with are given such that both are differentiable in the ordinary sense (we call these differentiable curves initialized at ). Then and are said to be equivalent at if and only if the derivatives of and at coincide. This defines an equivalence relation on the set of all differentiable curves initialized at , and equivalence classes of such curves are known as tangent vectors of at . The equivalence class of any such curve is denoted by . The tangent space of at , denoted by , is then defined as the set of all tangent vectors at ; it does not depend on the choice of coordinate chart . To define vector-space operations on , we use a chart and define a map by where . The map turns out to be bijective and may be used to transfer the vector-space operations on over