*For a more general — but much more technical — treatment of tangent vectors, see tangent space.*

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 **R**^{n}. 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 , where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter *t*.^{[1]} The unit tangent vector is given by

#### Example

Given the curve

in , the unit tangent vector at is given by

### Contravariance

If is given parametrically in the *n*-dimensional coordinate system *x ^{i}* (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 *u ^{i}*-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.^{[2]}

## 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^{[3]} 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

## References

## Bibliography

- Gray, Alfred (1993),
*Modern Differential Geometry of Curves and Surfaces*, Boca Raton: CRC Press. - Stewart, James (2001),
*Calculus: Concepts and Contexts*, Australia: Thomson/Brooks/Cole. - Kay, David (1988),
*Schaums Outline of Theory and Problems of Tensor Calculus*, New York: McGraw-Hill.