Geometric interpretation of a vector derivative

If $\bold{r}$ is the vector joining the origin $O$ of a coordinate system and the point $(x, y, z)$ , then specification of the vector function $\bold{r}(u)$ defines $x$ , $y$ and $z$ as function of $u$ . As $u$ changes, the terminal point of $\bold{r}$ describes a space curve having parametric equations $x = x(u)$ , $y = y(u)$ , $z = z(u)$ . If the parameter $u$ is the arc length $s$ measured from some fixed point on the curve, then
$\displaystyle \frac{d\bold{r}}{ds} = \bold{T}\cdots(9)$
is a unit vector in the direction of the tangent to the curve and is called the unit tangent vector. If $u$ is the time $t$ , then
$\displaystyle \frac{d\bold{r}}{dt} = \bold{v}\cdots(10)$
is the velocity with which the terminal point of $\bold{r}$ describes the curve. We have
$\displaystyle \bold{v} = \frac{d\bold{r}}{dt} = \frac{d\bold{r}}{ds}\frac{ds}{dt} = \frac{ds}{dt}\bold{T} = v\bold{T}\cdots(11)$
from which we see that the magnitude of $\bold{v}$ , often called the speed, is $v = ds/dt$ . Similarly,
$\displaystyle \frac{d^2\bold{r}}{dt^2} = \bold{a}\cdots(12)$
is the acceleration with which the terminal point of $\bold{r}$ describes the curve. These concepts have important applications in mechanics.