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 x is a linear derivation of the algebra defined by the set of germs at x.

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 t. The unit tangent vector is given by

Example

Given the curve in \R^3, the unit tangent vector at t = 0 is given by

Contravariance

If is given parametrically in the n-dimensional coordinate system xi (here we have used superscripts as an index instead of the usual subscript) by or then the tangent vector field is given by Under a change of coordinates the tangent vector in the ui -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 \mathbf{v} be a vector in \R^n. We define the directional derivative in the \mathbf{v} direction at a point by The tangent vector at the point \mathbf{x} may then be defined as

Properties

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

Tangent vector on manifolds

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