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
![{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{|\mathbf {r} '(t)|}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d82cc4377c3adb02a9c61f9b70145ebf83a80114)
Example
Given the curve
![{\displaystyle \mathbf {r} (t)=\left\{\left(1+t^{2},e^{2t},\cos {t}\right)\mid t\in \mathbb {R} \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8d3a9d19d2ebb454ac55f58b7378b17a93a2ec)
in
![{\displaystyle \mathbb {R} ^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
, the unit tangent vector at
![{\displaystyle t=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c)
is given by
![{\displaystyle \mathbf {T} (0)={\frac {\mathbf {r} '(0)}{\|\mathbf {r} '(0)\|}}=\left.{\frac {(2t,2e^{2t},-\sin {t})}{\sqrt {4t^{2}+4e^{4t}+\sin ^{2}{t}}}}\right|_{t=0}=(0,1,0)\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01e34eb0099b311d6a2cfd4ab2f69caa5e23f1a1)
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
![{\displaystyle \mathbf {r} =x^{i}=x^{i}(t),\quad a\leq t\leq b\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/195f9ba9cb230ac59c35ee4e38ed67a3ffebc7fe)
then the tangent vector field
![{\displaystyle \mathbf {T} =T^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/015fae9973b520daf15e51114340d0f3608e6af8)
is given by
![{\displaystyle T^{i}={\frac {dx^{i}}{dt}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4022ff5498a4e0b09d1a618e47d0bd94e2e5b3)
Under a change of coordinates
![{\displaystyle u^{i}=u^{i}(x^{1},x^{2},\ldots ,x^{n}),\quad 1\leq i\leq n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/241a3274b2da9e69f58159991382a3e2bf9c6b2f)
the tangent vector
![{\displaystyle {\bar {\mathbf {T} }}={\bar {T}}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/392a2db32c102c44cd2a395b953713e8a1576e98)
in the
ui-coordinate system is given by
![{\displaystyle {\bar {T}}^{i}={\frac {du^{i}}{dt}}={\frac {\partial u^{i}}{\partial x^{s}}}{\frac {dx^{s}}{dt}}=T^{s}{\frac {\partial u^{i}}{\partial x^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad44718e564968619c4fd2eed5a1d541755e060)
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
![{\displaystyle \nabla _{\mathbf {v} }f(\mathbf {x} )=\left.{\frac {d}{dt}}f(\mathbf {x} +t\mathbf {v} )\right|_{t=0}=\sum _{i=1}^{n}v_{i}{\frac {\partial f}{\partial x_{i}}}(\mathbf {x} )\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d55798d892fc890821f7f29ff6099982fbfaa9d)
The tangent vector at the point
![{\displaystyle \mathbf {x} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2)
may then be defined
[3] as
![{\displaystyle \mathbf {v} (f(\mathbf {x} ))\equiv (\nabla _{\mathbf {v} }(f))(\mathbf {x} )\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1da1a379742dc0e09d05c0a2859f1fd39ca7df)
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.