Vector tangent to a curve or surface at a given point
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
provided it exists and provided
, 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
![{\displaystyle (a\mathbf {v} +b\mathbf {w} )(f)=a\mathbf {v} (f)+b\mathbf {w} (f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79bcdd3faa59457bef107af75c66ccbcf31c533)
![{\displaystyle \mathbf {v} (af+bg)=a\mathbf {v} (f)+b\mathbf {v} (g)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ddfd8a12f0a8032c2f815b7c20fcc4fd100559)
![{\displaystyle \mathbf {v} (fg)=f(\mathbf {x} )\mathbf {v} (g)+g(\mathbf {x} )\mathbf {v} (f)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/194b8c6c2ced24995d407ab106e28e4a97193527)
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
![{\displaystyle D_{v}(af+bg)=aD_{v}(f)+bD_{v}(g)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10de26396ddf9af2b345d6e6027e315389a060e8)
Note that the derivation will by definition have the Leibniz property
![{\displaystyle D_{v}(f\cdot g)(x)=D_{v}(f)(x)\cdot g(x)+f(x)\cdot D_{v}(g)(x)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c1c076e866bb140819c1cd1e1580e9bdbfad68)
See also
References
- ^ J. Stewart (2001)
- ^ D. Kay (1988)
- ^ 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.