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 <math>x</math> is a linear derivation of the algebra defined by the set of germs at <math>x</math>.
MotivationEdit
Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.
CalculusEdit
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 Template:Mvar.<ref>J. Stewart (2001)</ref> The unit tangent vector is given by <math display="block">\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\,.</math>
ExampleEdit
Given the curve <math display="block">\mathbf{r}(t) = \left\{\left(1+t^2, e^{2t}, \cos{t}\right) \mid t\in\R\right\}</math> in <math>\R^3</math>, the unit tangent vector at <math>t = 0</math> is given by <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>
ContravarianceEdit
If <math>\mathbf{r}(t)</math> is given parametrically in the n-dimensional coordinate system Template:Math (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 <math display="block">\mathbf{r} = x^i = x^i(t), \quad a\leq t\leq b\,,</math> then the tangent vector field <math>\mathbf{T} = T^i</math> is given by <math display="block">T^i = \frac{dx^i}{dt}\,.</math> Under a change of coordinates <math display="block">u^i = u^i(x^1, x^2, \ldots, x^n), \quad 1\leq i\leq n</math> the tangent vector <math>\bar{\mathbf{T}} = \bar{T}^i</math> in the Template:Math-coordinate system is given by <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> 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.<ref>D. Kay (1988)</ref>
DefinitionEdit
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 <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> The tangent vector at the point <math>\mathbf{x}</math> may then be defined<ref>A. Gray (1993)</ref> as <math display="block">\mathbf{v}(f(\mathbf{x})) \equiv (\nabla_\mathbf{v}(f)) (\mathbf{x})\,.</math>
PropertiesEdit
Let <math>f,g:\mathbb{R}^n\to\mathbb{R}</math> be differentiable functions, let <math>\mathbf{v},\mathbf{w}</math> be tangent vectors in <math>\mathbb{R}^n</math> at <math>\mathbf{x}\in\mathbb{R}^n</math>, and let <math>a,b\in\mathbb{R}</math>. Then
- <math>(a\mathbf{v}+b\mathbf{w})(f)=a\mathbf{v}(f)+b\mathbf{w}(f)</math>
- <math>\mathbf{v}(af+bg)=a\mathbf{v}(f)+b\mathbf{v}(g)</math>
- <math>\mathbf{v}(fg)=f(\mathbf{x})\mathbf{v}(g)+g(\mathbf{x})\mathbf{v}(f)\,.</math>
Tangent vector on manifoldsEdit
Let <math>M</math> be a differentiable manifold and let <math>A(M)</math> be the algebra of real-valued differentiable functions on <math>M</math>. Then the tangent vector to <math>M</math> at a point <math>x</math> in the manifold is given by the derivation <math>D_v:A(M)\rightarrow\mathbb{R}</math> which shall be linear — i.e., for any <math>f,g\in A(M)</math> and <math>a,b\in\mathbb{R}</math> we have
- <math>D_v(af+bg)=aD_v(f)+bD_v(g)\,.</math>
Note that the derivation will by definition have the Leibniz property
- <math>D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x)\,.</math>
See alsoEdit
ReferencesEdit
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