Derivatives of Vector Functions

First Principles Definition

If $\vec{r}(t)=⟨x(t),y(t),z(t)⟩$ is a vector valued function with parameter $t$, then the derivative of $r(t)$ is:

$$\underset{\Delta t\to 0}{\lim }\frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t}$$

provided that it exists.

Practical Usage

Practically, we can just use:

$$\begin{array}{rl}{\vec{r}}^{\prime }(t)& =⟨f^{\prime }(t),g^{\prime }(t),h^{\prime }(t)⟩\\ & =f^{\prime }(t)\vec{i}+g^{\prime }(t)\vec{j}+h^{\prime }(t)\vec{k}\end{array}$$

This is also denoted by $\frac{d\vec{r}}{dt}$ and $\vec{T}(t)$.

Properties

Chain rule:

$$\begin{array}{rl}\frac{d}{dt}(f(t)\vec{u}(t))& =f^{\prime }(t)\vec{u}(t)+f(t){\vec{u}}^{\prime }(t)\\ & \\ \text{or}& \\ & \\ \frac{d}{dt}(f\vec{v})& =\frac{df}{dt}\vec{v}+f\frac{d\vec{v}}{dt}\end{array}$$

Dot product and cross product:

$$\begin{array}{rl}\frac{d}{dt}(\vec{u}\cdot \vec{v})& ={\vec{u}}^{\prime }\cdot \vec{v}+\vec{u}\cdot {\vec{v}}^{\prime }\\ \frac{d}{dt}(\vec{u}\times \vec{v})& ={\vec{u}}^{\prime }\times \vec{v}+\vec{u}\times {\vec{v}}^{\prime }\end{array}$$