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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 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 .

MotivationEdit

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

CalculusEdit

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

 

ExampleEdit

Given the curve

 

in  , the unit tangent vector at   is given by

 

ContravarianceEdit

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]

DefinitionEdit

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

 

PropertiesEdit

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

  1.  
  2.  
  3.  

Tangent vector on manifoldsEdit

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

 

ReferencesEdit

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

BibliographyEdit

  • 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.