Vector-valued function

A vector-valued function of one real variable assigns a vector (a point in 3D space) to each input. Written

$\mathbf{r}(t)=⟨x(t),y(t),z(t)⟩=x(t)\hat{\mathbf{i}}+y(t)\hat{\mathbf{j}}+z(t)\hat{\mathbf{k}}.$

Each coordinate is an ordinary scalar function of $t$. A vector-valued function is three scalar functions packaged together, and everything we know how to do with scalar functions — take limits, differentiate, integrate — applies componentwise.

The image of $\mathbf{r}$ — the set of points traced out as $t$ varies — is a curve in 3D space. The curve lives in 3D (you can walk around it, view it from different angles), but it has only one parameter pinning down where you are on it. One degree of freedom.

Don’t confuse degrees of freedom with the dimension of the ambient space. The curve is “one-dimensional” in the sense that one parameter labels its points, but it doesn’t sit in a 1D space — it sits in ${\mathbb{R}}^3$. The same distinction matters for surfaces (two degrees of freedom, but still in 3D).

Standard parameterizations

Line through ${\mathbf{P}}_0=(x_0,y_0,z_0)$ in direction $\mathbf{v}=⟨a,b,c⟩$:

$\mathbf{r}(t)=⟨x_0+at,y_0+bt,z_0+ct⟩.$

At $t=0$ you’re at ${\mathbf{P}}_0$; for each unit increase in $t$, you move by $\mathbf{v}$. If $|\mathbf{v}|=1$, $t$ measures arc length directly.

Circle of radius $R$ in the $xy$-plane, counterclockwise:

$\mathbf{r}(t)=⟨R\cos t,R\sin t,0⟩,0\le t\le 2\pi .$

Helix:

$\mathbf{r}(t)=⟨\cos t,\sin t,t⟩.$

The $(x,y)$ projection traces a unit circle while $z$ climbs steadily. A corkscrew that rises $2\pi$ for every revolution.

Graph $y=f(x)$: use $x$ as the parameter, $\mathbf{r}(t)=⟨t,f(t),0⟩$.

Same curve, different parameterizations

${\mathbf{r}}_1(t)=⟨t,t^2⟩$ for $0\le t\le 1$ traces the parabola from $(0,0)$ to $(1,1)$. ${\mathbf{r}}_2(t)=⟨1-t,(1-t{)}^2⟩$ traces the same parabola in reverse. Same image, different orientation. For line integrals, orientation sometimes matters (vector line integrals flip sign) and sometimes doesn’t (scalar line integrals are orientation-independent).

Domain, limit, continuity

The domain of $\mathbf{r}$ is the intersection of the domains of its component functions.

The limit is componentwise (provided each component limit exists):

${\lim }_{t\to a}\mathbf{r}(t)=⟨{\lim }_{t\to a}x(t),{\lim }_{t\to a}y(t),{\lim }_{t\to a}z(t)⟩.$

$\mathbf{r}$ is continuous at $a$ iff each component is continuous at $a$.

Looking ahead

The next steps — differentiation, tangent vector, Arc length — develop the calculus of curves and lead directly to line integrals in vector fields.