Editing Tangent space (section)

== Informal description ==

[[Image:Image Tangent-plane.svg|thumb|A pictorial representation of the tangent space of a single point <math> x </math> on a [[sphere]]. A vector in this tangent space represents a possible velocity (of something moving on the sphere) at <math> x </math>. After moving in that direction to a nearby point, the velocity would then be given by a vector in the tangent space of that point—a different tangent space that is not shown.]]

In [[differential geometry]], one can attach to every point <math> x </math> of a [[differentiable manifold]] a ''tangent space''—a real [[vector space]] that intuitively contains the possible directions in which one can tangentially pass through <math> x </math>. The elements of the tangent space at <math> x </math> are called the ''[[tangent vector]]s'' at <math> x </math>. This is a generalization of the notion of a [[Vector (mathematics and physics)|vector]], based at a given initial point, in a [[Euclidean space]]. The [[dimension of a vector space|dimension]] of the tangent space at every point of a [[connected space|connected]] manifold is the same as that of the [[manifold]] itself.

For example, if the given manifold is a <math> 2 </math>-[[sphere]], then one can picture the tangent space at a point as the plane that touches the sphere at that point and is [[perpendicular]] to the sphere's radius through the point. More generally, if a given manifold is thought of as an [[embedding|embedded]] [[submanifold]] of [[Euclidean space]], then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining [[parallel transport]]. Many authors in [[differential geometry]] and [[general relativity]] use it.<ref>{{cite book |last=do Carmo |first=Manfredo P. |title=Differential Geometry of Curves and Surfaces|year=1976 |publisher=Prentice-Hall }}: </ref><ref>{{cite book |last=Dirac |first=Paul A. M. |title=General Theory of Relativity |orig-year=1975 |year=1996 |publisher=Princeton University Press |isbn=0-691-01146-X }}</ref> More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.

In [[algebraic geometry]], in contrast, there is an intrinsic definition of the ''tangent space at a point'' of an [[algebraic variety]] <math> V </math> that gives a vector space with dimension at least that of <math> V </math> itself. The points <math> p </math> at which the dimension of the tangent space is exactly that of <math> V </math> are called ''non-singular'' points; the others are called ''singular'' points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of <math> V </math> are those where the "test to be a manifold" fails. See [[Zariski tangent space]].

Once the tangent spaces of a manifold have been introduced, one can define [[vector field]]s, which are abstractions of the velocity field of particles moving in space. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such 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.