Vector Functions
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 3.1: “Vector-Valued Functions and Space Curves” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/3-1-vector-valued-functions-and-space-curves |
| Textbook used in class | Stewart, Calculus, Section 13.1: “Vector Functions and Space Curves” (Examples 1, 2, 3) |
Opening Scenario
A parametric curve in 2D is described by $x = f(t)$, $y = g(t)$: two scalar functions of $t$. A vector function bundles those into one object: $\mathbf{r}(t) = \langle f(t), g(t)\rangle$. In 3D, add a third component: $\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle$. At each $t$, the output is a vector; as $t$ varies, the tip of that vector traces a curve in space.
Quick Reference
Vector-valued function: $\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle = f(t)\,\mathbf{i} + g(t)\,\mathbf{j} + h(t)\,\mathbf{k}$.
Domain: the set of $t$-values where all three component functions are defined.
Limit: \[ \lim_{t\to a}\mathbf{r}(t) = \left\langle \lim_{t\to a}f(t),\; \lim_{t\to a}g(t),\; \lim_{t\to a}h(t)\right\rangle, \] provided all three component limits exist.
Continuity: $\mathbf{r}$ is continuous at $a$ if $\lim_{t\to a}\mathbf{r}(t) = \mathbf{r}(a)$, i.e., each component is continuous at $a$.
Key Concepts
1. Evaluating a Vector Function
Example 1. For $\mathbf{r}(t) = \langle e^t, \cos t, t\ln t\rangle$, find the domain and $\mathbf{r}(1)$. (Stewart 13.1, Example 1.)
Domain: $e^t$ and $\cos t$ are defined for all $t$; $t\ln t$ requires $t > 0$. Domain: $(0, \infty)$.
$\mathbf{r}(1) = \langle e, \cos 1, 0\rangle$.
2. Limit of a Vector Function
Example 2. Find $\displaystyle\lim_{t\to 0}\left\langle \frac{\sin t}{t},\; t^2 - 1,\; e^t\right\rangle$.
$= \left\langle 1,\; -1,\; 1\right\rangle$.
(Each component limit is computed separately; the limit of $\sin t/t$ as $t\to 0$ is $1$.)
3. Vector Functions as Parametric Equations
A vector function $\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle$ gives parametric equations $x = f(t)$, $y = g(t)$, $z = h(t)$. The curve traced is the same object; the vector function is a compact notation.
evaluating the limit of a vector function by “plugging in” before checking each component. The limit of $\mathbf{r}(t)$ as $t\to a$ exists only when all three component limits exist and are finite. If any component limit fails (say, $f(t) \to \infty$), the vector limit does not exist. Always check each component separately before combining.
Common Misconceptions
the limit of a vector function exists whenever the function is defined at the approach point.
This is the limit-equals-function-value error extended to vector functions. The limit $\lim_{t\to a}\mathbf{r}(t)$ exists only when all three component limits exist separately and are finite. Even if $\mathbf{r}(a)$ is defined, the limit could fail in one component. For example, if $f(t) \to \infty$ as $t \to a$ while $g(t)$ and $h(t)$ converge, the vector limit does not exist. Each component must be verified independently.
Leveled Practice
Problem 1. For $\mathbf{r}(t) = \langle t^2, \sqrt{t}, \ln(1-t)\rangle$, state the domain.
Show answer
$t^2$: all $t$. $\sqrt{t}$: $t \geq 0$. $\ln(1-t)$: $1 - t > 0$, so $t < 1$. Domain: $[0, 1)$.