Informal Description
In differential geometry, one can attach to every point x of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible "directions" at which one can tangentially pass through x. The elements of the tangent space are called tangent vectors at x. This is a generalization of the notion of a bound vector in a Euclidean space. All the tangent spaces have the same dimension, equal to the dimension of the manifold.
For example, if the given manifold is a 2-sphere, one can picture the tangent space at a point as the plane which 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 embedded submanifold of Euclidean space one can picture the tangent space in this literal fashion.
In algebraic geometry, in contrast, there is an intrinsic definition of tangent space at a point P of a variety V, that gives a vector space of dimension at least that of V. The points P at which the dimension is exactly that of V are called the non-singular points; the others are singular points. For example, a curve that crosses itself doesn't have a unique tangent line at that point. The singular points of V are those where the 'test to be a manifold' fails. See Zariski tangent space.
Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. 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 can be "glued together" to form a new differentiable manifold of twice the dimension of the original manifold, called the tangent bundle of the manifold.
Read more about this topic: Tangent Space
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