Vector-valued function

Vector-valued functions are functions of the form:

$$\vec{r}(t)=⟨f(t),g(t),h(t)⟩=f(t)\hat{i}+g(t)\hat{j}+h(t)\hat{k}$$

where $f$, $g$, and $h$ are real-valued functions with the parameter $t$.

Calculations

Standard limits and calculus applies. Where we apply some operation to $\vec{r}(t)$, we apply term-by-term for each function component. Important of note is that the product rule between a real-valued function and vector-valued function applies.

$$(u(t)\vec{p}(t){)}^{\prime }=u^{\prime }(t)\vec{p}(t)+u(t){\vec{p}}^{\prime }(t)$$

As does the dot product:

$$(\vec{r}(t)\cdot \vec{p}(t){)}^{\prime }={\vec{r}}^{\prime }(t)\cdot \vec{p}(t)+\vec{r}(t)\cdot {\vec{p}}^{\prime }(t)$$

Key concepts

• Arc length
• Normal-tangential coordinates