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Tangent vector

From Wikipedia, the free encyclopedia

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 , 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 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]

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

See also

References

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

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.
This page was last edited on 30 August 2021, at 10:26
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