13.1 Vector Functions and Space Curves

A function asigns a domain to a range. A vector-valued function, or "vector function" is a function whose domain is a set of real numbers an whose range is a set of vectors. $f(t)$, $g(t)$, and $h(t)$ are components of vector $\vec{r}(t)$. Therefore, $f$, $g$, and $h$ are real valued functions called the "components of $\vec{r}$". Writen as $\vec{r}(t)=<f(t), g(t), h(t)> = f(t)\hat{i}+g(t)\hat{j}+h(t)\hat{k}$. $t$ is an indepdendent variable and often represents time.

Limit of Vector Functions

The limit of function $\vec{r}$ is defined by taking the limits of its components: $\lim_{t \to a}\vec{r} = <\lim_{t \to a}f(t),\lim_{t \to a}g(t), \lim_{t \to a}h(t)>$ provided the limit of the components exist.

Continuity of Vector Functions

Vector function $\vec{r}$ is continuous at $a$ if: $\lim_{t \to a} \vec{r}(t) = \vec{r}(a)$ meaning $\vec{r}$ is continuous if its component functions $f$, $g$, and $h$ are continuous at $t=a$.

Space Curves