Tangent vector

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 ${\displaystyle {\mathbf{r}}^{\prime }(t)}$ provided it exists and provided ${\displaystyle {\mathbf{r}}^{\prime }(t)\ne \mathbf{0}}$, 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 $${\displaystyle \mathbf{T}(t)=\frac{{\mathbf{r}}^{\prime }(t)}{|{\mathbf{r}}^{\prime }(t)|}.}$$

Example

Given the curve $${\displaystyle \mathbf{r}(t)=\left\{\left(1+t^2,e^{2t},\cos t\right)\mid t\in \mathbb{R}\right\}}$$ in ${\displaystyle {\mathbb{R}}^3}$, the unit tangent vector at ${\displaystyle t=0}$ is given by $${\displaystyle \mathbf{T}(0)=\frac{{\mathbf{r}}^{\prime }(0)}{\Vert {\mathbf{r}}^{\prime }(0)\Vert }={\frac{(2t,2e^{2t},-\sin t)}{\sqrt{4t^2+4e^{4t}+{\sin }^{2}t}}|}_{t=0}=(0,1,0).}$$ Where the components of the tangent vector are found by taking the derivative of each corresponding component of the curve with respect to ${\displaystyle t}$.

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 ${\displaystyle \mathbf{r}(t)=(x^1(t),x^2(t),\dots ,x^n(t))}$ or $${\displaystyle \mathbf{r}=x^i=x^i(t),a\le t\le b,}$$ then the tangent vector field ${\displaystyle \mathbf{T}=T^i}$ is given by $${\displaystyle T^i=\frac{d{x}^i}{dt}.}$$ Under a change of coordinates $${\displaystyle u^i=u^i(x^1,x^2,\dots ,x^n),1\le i\le n}$$ the tangent vector ${\displaystyle \overline{\mathbf{T}}={\overline{T}}^i}$ in the uⁱ-coordinate system is given by $${\displaystyle {\overline{T}}^i=\frac{d{u}^i}{dt}=\frac{\mathrm{\partial }{u}^i}{\mathrm{\partial }{x}^s}\frac{d{x}^s}{dt}=T^s\frac{\mathrm{\partial }{u}^i}{\mathrm{\partial }{x}^s}}$$ 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.