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 .

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


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


Let be a parametric smooth curve. The tangent vector is given by , where we have used the a prime instead of the usual dot to indicate differentiation with respect to parameter t.[1] The unit tangent vector is given by


Given the curve

in , the unit tangent vector at is given by


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.[2]


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


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


  1. J. Stewart (2001)
  2. D. Kay (1988)
  3. A. Gray (1993)


  • 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.
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