A tangent vector is a vector
, which is tangent
to a curve
or, more generally, a manifold
at a certain point. If you have a curve
c(t), parametrized by t, its tangent vector is just c'(t), where ' means
. A manifold, of course, generally has many tangent
vectors. In fact, it has a tangent vector space
at its each point.
Locally any manifold looks like its tangent vector space, but in general
not globally. More concretely, if you have a sphere S2
3-dimensional space, if you look at it closely enough, it looks flat, but
as you zoom farther away, the curvature
A tangent vector can be mathematically defined as a directional
derivative in some direction on the manifold. Tangent vectors are very
important in differential geometry.