# 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 $\displaystyle{ x }$ is a linear derivation of the algebra defined by the set of germs at $\displaystyle{ x }$.

## Motivation

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

### Calculus

Let $\displaystyle{ \mathbf{r}(t) }$ be a parametric smooth curve. The tangent vector is given by $\displaystyle{ \mathbf{r}'(t) }$ provided it exists and provided $\displaystyle{ \mathbf{r}'(t)\neq \mathbf{0} }$, 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 $\displaystyle{ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\,. }$

#### Example

Given the curve $\displaystyle{ \mathbf{r}(t) = \left\{\left(1+t^2, e^{2t}, \cos{t}\right) \mid t\in\R\right\} }$ in $\displaystyle{ \R^3 }$, the unit tangent vector at $\displaystyle{ t = 0 }$ is given by $\displaystyle{ \mathbf{T}(0) = \frac{\mathbf{r}'(0)}{\|\mathbf{r}'(0)\|} = \left.\frac{(2t, 2e^{2t}, -\sin{t})}{\sqrt{4t^2 + 4e^{4t} + \sin^2{t}}}\right|_{t=0} = (0,1,0)\,. }$

### Contravariance

If $\displaystyle{ \mathbf{r}(t) }$ is given parametrically in the n-dimensional coordinate system xi (here we have used superscripts as an index instead of the usual subscript) by $\displaystyle{ \mathbf{r}(t) = (x^1(t), x^2(t), \ldots, x^n(t)) }$ or $\displaystyle{ \mathbf{r} = x^i = x^i(t), \quad a\leq t\leq b\,, }$ then the tangent vector field $\displaystyle{ \mathbf{T} = T^i }$ is given by $\displaystyle{ T^i = \frac{dx^i}{dt}\,. }$ Under a change of coordinates $\displaystyle{ u^i = u^i(x^1, x^2, \ldots, x^n), \quad 1\leq i\leq n }$ the tangent vector $\displaystyle{ \bar{\mathbf{T}} = \bar{T}^i }$ in the ui-coordinate system is given by $\displaystyle{ \bar{T}^i = \frac{du^i}{dt} = \frac{\partial u^i}{\partial x^s} \frac{dx^s}{dt} = T^s \frac{\partial u^i}{\partial x^s} }$ 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 $\displaystyle{ f: \R^n \to \R }$ be a differentiable function and let $\displaystyle{ \mathbf{v} }$ be a vector in $\displaystyle{ \R^n }$. We define the directional derivative in the $\displaystyle{ \mathbf{v} }$ direction at a point $\displaystyle{ \mathbf{x} \in \R^n }$ by $\displaystyle{ \nabla_\mathbf{v} f(\mathbf{x}) = \left.\frac{d}{dt} f(\mathbf{x} + t\mathbf{v})\right|_{t=0} = \sum_{i=1}^{n} v_i \frac{\partial f}{\partial x_i}(\mathbf{x})\,. }$ The tangent vector at the point $\displaystyle{ \mathbf{x} }$ may then be defined[3] as $\displaystyle{ \mathbf{v}(f(\mathbf{x})) \equiv (\nabla_\mathbf{v}(f)) (\mathbf{x})\,. }$

## Properties

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

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

## Tangent vector on manifolds

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

$\displaystyle{ D_v(af+bg)=aD_v(f)+bD_v(g)\,. }$

Note that the derivation will by definition have the Leibniz property

$\displaystyle{ D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x)\,. }$