Tangent vector

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 $$x$$ is a linear derivation of the algebra defined by the set of germs at $$x$$.

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

Calculus
Let $$\mathbf{r}(t)$$ be a parametric smooth curve. The tangent vector is given by $$\mathbf{r}'(t)$$ provided it exists and provided $$\mathbf{r}'(t)\neq \mathbf{0}$$, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter $t$. The unit tangent vector is given by $$\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\,.$$

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

Contravariance
If $$\mathbf{r}(t)$$ is given parametrically in the n-dimensional coordinate system $x^{i}$ (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 $u^{i}$-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.

Definition
Let $$f: \R^n \to \R$$ be a differentiable function and let $$\mathbf{v}$$ be a vector in $$\R^n$$. We define the directional derivative in the $$\mathbf{v}$$ direction at a point $$\mathbf{x} \in \R^n$$ by $$\nabla_\mathbf{v} f(\mathbf{x}) = \left.\frac{d}{dt} f(\mathbf{x} + t\mathbf{v})\right|_{t=0} = \sum_{i=1}^{n} v_i \frac{\partial f}{\partial x_i}(\mathbf{x})\,.$$ The tangent vector at the point $$\mathbf{x}$$ may then be defined as $$\mathbf{v}(f(\mathbf{x})) \equiv (\nabla_\mathbf{v}(f)) (\mathbf{x})\,.$$

Properties
Let $$f,g:\mathbb{R}^n\to\mathbb{R}$$ 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
Let $$M$$ be a differentiable manifold and let $$A(M)$$ be the algebra of real-valued differentiable functions on $$M$$. 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 &mdash; 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 definition have the Leibniz property
 * $$D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x)\,.$$