Tangent vector

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For a more general, but much more technical, treatment of tangent vectors, see tangent space.

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. In other words, a tangent vector at the point x is a linear derivation of the algebra defined by the set of germs at x.

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 \mathbf{r}(t) be a parametric smooth curve. The tangent vector is given by \mathbf{r}^\prime(t), where we have used the a prime instead of the usual dot to indicate differentiation with respect to parameter t.[1] the unit tangent vector is given by

\mathbf{T}(t)=\frac{\mathbf{r}^\prime(t)}{|\mathbf{r}^\prime(t)|}\,.

Example[edit]

Given the curve

\mathbf{r}(t)=\{(1+t^2,e^t,\cos{t})|\ t\in\mathbb{R}\}

in \mathbb{R}^3, the unit tangent vector at time t=0 is given by

\mathbf{T}(0)=\frac{\mathbf{r}^\prime(0)}{|\mathbf{r}^\prime(0)|}=\left.\frac{(1+t^2,e^t,\cos{t})}{\sqrt{4t^2+e^{2t}+\sin^2{t}}}\right|_{t=0}=(0,1,0)\,.

Contravariance[edit]

If \mathbf{r}(t) is given parametrically in the n-dimensional coordinate system xi (here we have used superscripts as an index instead of the usual subscript) by \mathbf{r}(t)=(x^1(t),x^2(t),\ldots,x^n(t)) or

\mathbf{r}=x^i=x^i(t),\quad a\leq t\leq b\,,

then the tangent vector field \mathbf{T}=T^i is given by

T^i=\frac{dx^i}{dt}\,.

Under a change of coordinates

u^i=u^i(x^1,x^2,\ldots,x^n),\quad 1\leq i\leq n

the tangent vector \bar{\mathbf{T}}=\bar{T}^i in the ui-coordinate system is given by

\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}

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 f:\mathbb{R}^n\rightarrow\mathbb{R} be a differentiable function and let \mathbf{v} be a vector in \mathbb{R}^n. We define the directional derivative in the \mathbf{v} direction at a point \mathbf{x}\in\mathbb{R}^n by

D_\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})\,.

The tangent vector at the point \mathbf{x} may then be defined[3] as

\mathbf{v}(f(\mathbf{x}))\equiv D_\mathbf{v}(f(\mathbf{x}))\,.

Properties[edit]

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

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

Tangent Vector on Manifolds[edit]

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

D_v(af+bg)=aD_v(f)+bD_v(g)\,.

Note that the derivation will by defintion have the Leibniz property

D_v(f\cdot g)=D_v(f)\cdot g(x)+f(x)\cdot D_v(g)\,.

References[edit]

  1. ^ J. Stewart (2001)
  2. ^ D. Kay (1988)
  3. ^ A. Gray (1993)

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 .