Tangent vector: Difference between revisions
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{{short description|Vector tangent to a curve or surface at a given point}} |
{{short description|Vector tangent to a curve or surface at a given point}} |
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{{For|a more general, but more technical, treatment of tangent vectors|Tangent space}} |
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In [[mathematics]], a '''tangent vector''' is a [[Vector (geometry)|vector]] that is [[tangent]] to a [[curve]] or [[Surface (mathematics)|surface]] at a given point. Tangent vectors are described in the [[differential geometry of curves]] in the context of curves in '''R'''<sup>''n''</sup>. More generally, tangent vectors are elements of a [[tangent space]] of a [[differentiable manifold]]. Tangent vectors can also be described in terms of [[Germ (mathematics)|germs]]. Formally, a tangent vector at the point <math>x</math> is a linear [[Derivation (differential algebra)|derivation]] of the algebra defined by the set of germs at <math>x</math>. |
In [[mathematics]], a '''tangent vector''' is a [[Vector (geometry)|vector]] that is [[tangent]] to a [[curve]] or [[Surface (mathematics)|surface]] at a given point. Tangent vectors are described in the [[differential geometry of curves]] in the context of curves in '''R'''<sup>''n''</sup>. More generally, tangent vectors are elements of a [[tangent space]] of a [[differentiable manifold]]. Tangent vectors can also be described in terms of [[Germ (mathematics)|germs]]. Formally, a tangent vector at the point <math>x</math> is a linear [[Derivation (differential algebra)|derivation]] of the algebra defined by the set of germs at <math>x</math>. |
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=== Calculus === |
=== Calculus === |
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Let <math>\mathbf{r}(t)</math> be a parametric [[smooth curve]]. The tangent vector is given by <math>\mathbf{r} |
Let <math>\mathbf{r}(t)</math> be a parametric [[smooth curve]]. The tangent vector is given by <math>\mathbf{r}'(t)</math> provided it exists and provided <math>\mathbf{r}'(t)\neq \mathbf{0}</math>, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter {{mvar|t}}.<ref>J. Stewart (2001)</ref> The unit tangent vector is given by |
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<math display="block">\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\,.</math> |
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==== Example ==== |
==== Example ==== |
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Given the curve |
Given the curve |
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<math display="block">\mathbf{r}(t) = \left\{\left(1+t^2, e^{2t}, \cos{t}\right) \mid t\in\R\right\}</math> |
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in <math>\ |
in <math>\R^3</math>, the unit tangent vector at <math>t = 0</math> is given by |
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<math display="block">\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)\,.</math> |
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=== Contravariance === |
=== Contravariance === |
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If <math>\mathbf{r}(t)</math> is given parametrically in the [[n-dimensional coordinate system|''n''-dimensional coordinate system]] ''x<sup>i</sup>'' (here we have used superscripts as an index instead of the usual subscript) by <math>\mathbf{r}(t)=(x^1(t),x^2(t),\ldots,x^n(t))</math> or |
If <math>\mathbf{r}(t)</math> is given parametrically in the [[n-dimensional coordinate system|''n''-dimensional coordinate system]] {{math|''x<sup>i</sup>''}} (here we have used superscripts as an index instead of the usual subscript) by <math>\mathbf{r}(t) = (x^1(t), x^2(t), \ldots, x^n(t))</math> or |
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<math display="block">\mathbf{r} = x^i = x^i(t), \quad a\leq t\leq b\,,</math> |
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then the tangent vector field <math>\mathbf{T}=T^i</math> is given by |
then the tangent vector field <math>\mathbf{T} = T^i</math> is given by |
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<math display="block">T^i = \frac{dx^i}{dt}\,.</math> |
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Under a change of coordinates |
Under a change of coordinates |
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<math display="block">u^i = u^i(x^1, x^2, \ldots, x^n), \quad 1\leq i\leq n</math> |
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the tangent vector <math>\bar{\mathbf{T}}=\bar{T}^i</math> in the ''u<sup>i</sup>''-coordinate system is given by |
the tangent vector <math>\bar{\mathbf{T}} = \bar{T}^i</math> in the {{math|''u<sup>i</sup>''}}-coordinate system is given by |
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<math display="block">\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}</math> |
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where we have used the [[Einstein notation|Einstein summation convention]]. Therefore, a tangent vector of a smooth curve will transform as a [[Covariance and contravariance of vectors|contravariant]] tensor of order one under a change of coordinates.<ref>D. Kay (1988)</ref> |
where we have used the [[Einstein notation|Einstein summation convention]]. Therefore, a tangent vector of a smooth curve will transform as a [[Covariance and contravariance of vectors|contravariant]] tensor of order one under a change of coordinates.<ref>D. Kay (1988)</ref> |
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== Definition == |
== Definition == |
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Let <math>f:\ |
Let <math>f: \R^n \to \R</math> be a differentiable function and let <math>\mathbf{v}</math> be a vector in <math>\R^n</math>. We define the directional derivative in the <math>\mathbf{v}</math> direction at a point <math>\mathbf{x} \in \R^n</math> by |
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<math display="block">\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})\,.</math> |
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The tangent vector at the point <math>\mathbf{x}</math> may then be defined<ref>A. Gray (1993)</ref> as |
The tangent vector at the point <math>\mathbf{x}</math> may then be defined<ref>A. Gray (1993)</ref> as |
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<math display="block">\mathbf{v}(f(\mathbf{x})) \equiv (\nabla_\mathbf{v}(f)) (\mathbf{x})\,.</math> |
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== Properties == |
== Properties == |
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Note that the derivation will by definition have the Leibniz property |
Note that the derivation will by definition have the Leibniz property |
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:<math>D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x)\,.</math> |
:<math>D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x)\,.</math> |
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== See also == |
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*{{slink|Differentiable curve#Tangent vector}} |
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*{{slink|Differentiable surface#Tangent plane and normal vector}} |
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== References == |
== References == |
Latest revision as of 08:31, 1 December 2023
In mathematics, a tangent vector is a vector that is tangent to a curve oder 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
[edit]Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.
Calculus
[edit]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
[edit]Given the curve in , the unit tangent vector at is given by
Contravariance
[edit]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 oder 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
[edit]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
[edit]Let be differentiable functions, let be tangent vectors in at , and let . Then
Tangent vector on manifolds
[edit]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
[edit]References
[edit]Bibliography
[edit]- 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.