Tangent vector

Short description: 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 [math]\displaystyle{ x }[/math] is a linear derivation of the algebra defined by the set of germs at [math]\displaystyle{ x }[/math].

Motivation

Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.

Calculus

Let [math]\displaystyle{ \mathbf{r}(t) }[/math] be a parametric smooth curve. The tangent vector is given by [math]\displaystyle{ \mathbf{r}'(t) }[/math] provided it exists and provided [math]\displaystyle{ \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 t. The unit tangent vector is given by [math]\displaystyle{ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\,. }[/math]

Example

Given the curve [math]\displaystyle{ \mathbf{r}(t) = \left\{\left(1+t^2, e^{2t}, \cos{t}\right) \mid t\in\R\right\} }[/math] in [math]\displaystyle{ \R^3 }[/math], the unit tangent vector at [math]\displaystyle{ t = 0 }[/math] is given by [math]\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)\,. }[/math]

Contravariance

If [math]\displaystyle{ \mathbf{r}(t) }[/math] is given parametrically in the n-dimensional coordinate system xⁱ (here we have used superscripts as an index instead of the usual subscript) by [math]\displaystyle{ \mathbf{r}(t) = (x^1(t), x^2(t), \ldots, x^n(t)) }[/math] or [math]\displaystyle{ \mathbf{r} = x^i = x^i(t), \quad a\leq t\leq b\,, }[/math] then the tangent vector field [math]\displaystyle{ \mathbf{T} = T^i }[/math] is given by [math]\displaystyle{ T^i = \frac{dx^i}{dt}\,. }[/math] Under a change of coordinates [math]\displaystyle{ u^i = u^i(x^1, x^2, \ldots, x^n), \quad 1\leq i\leq n }[/math] the tangent vector [math]\displaystyle{ \bar{\mathbf{T}} = \bar{T}^i }[/math] in the uⁱ-coordinate system is given by [math]\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} }[/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.

Definition

Let [math]\displaystyle{ f: \R^n \to \R }[/math] be a differentiable function and let [math]\displaystyle{ \mathbf{v} }[/math] be a vector in [math]\displaystyle{ \R^n }[/math]. We define the directional derivative in the [math]\displaystyle{ \mathbf{v} }[/math] direction at a point [math]\displaystyle{ \mathbf{x} \in \R^n }[/math] by [math]\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})\,. }[/math] The tangent vector at the point [math]\displaystyle{ \mathbf{x} }[/math] may then be defined as [math]\displaystyle{ \mathbf{v}(f(\mathbf{x})) \equiv (\nabla_\mathbf{v}(f)) (\mathbf{x})\,. }[/math]

Properties

Let [math]\displaystyle{ f,g:\mathbb{R}^n\to\mathbb{R} }[/math] be differentiable functions, let [math]\displaystyle{ \mathbf{v},\mathbf{w} }[/math] be tangent vectors in [math]\displaystyle{ \mathbb{R}^n }[/math] at [math]\displaystyle{ \mathbf{x}\in\mathbb{R}^n }[/math], and let [math]\displaystyle{ a,b\in\mathbb{R} }[/math]. Then

1. [math]\displaystyle{ (a\mathbf{v}+b\mathbf{w})(f)=a\mathbf{v}(f)+b\mathbf{w}(f) }[/math]
2. [math]\displaystyle{ \mathbf{v}(af+bg)=a\mathbf{v}(f)+b\mathbf{v}(g) }[/math]
3. [math]\displaystyle{ \mathbf{v}(fg)=f(\mathbf{x})\mathbf{v}(g)+g(\mathbf{x})\mathbf{v}(f)\,. }[/math]

Tangent vector on manifolds

Let [math]\displaystyle{ M }[/math] be a differentiable manifold and let [math]\displaystyle{ A(M) }[/math] be the algebra of real-valued differentiable functions on [math]\displaystyle{ M }[/math]. Then the tangent vector to [math]\displaystyle{ M }[/math] at a point [math]\displaystyle{ x }[/math] in the manifold is given by the derivation [math]\displaystyle{ D_v:A(M)\rightarrow\mathbb{R} }[/math] which shall be linear — i.e., for any [math]\displaystyle{ f,g\in A(M) }[/math] and [math]\displaystyle{ a,b\in\mathbb{R} }[/math] we have

[math]\displaystyle{ D_v(af+bg)=aD_v(f)+bD_v(g)\,. }[/math]

Note that the derivation will by definition have the Leibniz property

[math]\displaystyle{ D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x)\,. }[/math]

See also

• Differentiable curve § Tangent vector
• Differentiable surface § Tangent plane and normal vector

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 .