In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. In other words, a tangent vector at the point
Contents
Motivation
Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.
Calculus
Let
Example
Given the curve
in
Contravariance
If
then the tangent vector field
Under a change of coordinates
the tangent vector
where we have used the Einstein summation convention. Therefore, a tangent vector of a smooth curve will transform as a contravariant tensor of order one under a change of coordinates.
Definition
Let
The tangent vector at the point
Properties
Let
-
( a v + b w ) ( f ) = a v ( f ) + b w ( f ) -
v ( a f + b g ) = a v ( f ) + b v ( g ) -
v ( f g ) = f ( x ) v ( g ) + g ( x ) v ( f ) .
Tangent vector on manifolds
Let
Note that the derivation will by definition have the Leibniz property