Tangent vector


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. Formally, a tangent vector at the point is a linear derivation of the algebra defined by the set of germs at.

Motivation

Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.

Calculus

Let be a parametric smooth curve. The tangent vector is given by provided it exists and provided, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter. The unit tangent vector is given by

Example

Given the curve
in, the unit tangent vector at is given by
Where the components of the tangent vector are found by taking the derivative of each corresponding component of the curve with respect to.

Contravariance

If is given parametrically in the n-dimensional coordinate system by or
then the tangent vector field is given by
Under a change of coordinates
the tangent vector in the -coordinate system is given by
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 be a differentiable function and let be a vector in. We define the directional derivative in the direction at a point by
The tangent vector at the point may then be defined as

Properties

Let be differentiable functions, let be tangent vectors in at, and let. Then

Tangent vector on manifolds

Let be a differentiable manifold and let be the algebra of real-valued differentiable functions on. Then the tangent vector to at a point in the manifold is given by the derivation which shall be linear — i.e., for any and we have
Note that the derivation will by definition have the Leibniz property