13.110 min read

Vector-Valued Functions

A vector-valued function $\mathbf{f}: D \subseteq \mathbb{R}^n \to \mathbb{R}^m$ maps each input vector $\mathbf{x} \in \mathbb{R}^n$ to an output vector $\mathbf{f}(\mathbf{x}) \in \mathbb{R}^m$ . When $m > 1$ , the output has multiple components, each depending on all inputs.

We can always write $\mathbf{f} = (f_1, f_2, \ldots, f_m)$ where each component function $f_i: \mathbb{R}^n \to \mathbb{R}$ is scalar-valued. For example, a function $\mathbf{f}: \mathbb{R}^2 \to \mathbb{R}^3$ might be written $\mathbf{f}(x,y) = (x^2+y, xy, e^y)$ .

Vector-valued functions generalize the notion of "function" in several directions at once:

$n=1$ , $m>1$ : parametric curve (trajectory through $\mathbb{R}^m$ as parameter varies)
$n>1$ , $m=1$ : scalar field (previously studied)
$n>1$ , $m>1$ : vector field or transformation (maps one space to another)

Formal View

Definition 13.1 — Vector-Valued Function

A vector-valued function is a map

\mathbf{f}: D \subseteq \mathbb{R}^n \to \mathbb{R}^m

defined by

m

component functions:

\mathbf{f}(\mathbf{x}) = \begin{bmatrix}f_1(\mathbf{x})\\\vdots\\f_m(\mathbf{x})\end{bmatrix}

where each

f_i: D \to \mathbb{R}

is a scalar-valued function.

$\mathbf{f}$ is continuous (resp. differentiable) iff each component $f_i$ is continuous (resp. differentiable).

Example 13.1

The function

\mathbf{f}(t) = (\cos t, \sin t, t)

maps

\mathbb{R} \to \mathbb{R}^3

and traces a helix. Its components are

f_1(t)=\cos t

f_2(t)=\sin t

f_3(t)=t

Why This Matters

Vector-valued functions appear whenever a single input determines multiple outputs — coordinate transformations, physical trajectories, and neural network layers are all examples.

Coordinate transformations: converting between Cartesian, polar, and spherical coordinates
Physics: position, velocity, and acceleration as vector-valued functions of time
Neural network layers: each layer is a vector-valued function of the previous layer's output

Learning Resources

Introduction to Vector-Valued Functions

Khan Academy

Introduction to vector-valued functions with parametric curve examples.

10 min

Vector Functions and Space Curves

MIT OpenCourseWare

MIT lecture on vector-valued functions and their geometric interpretation.

50 min

Quiz

Question 1

A function $\mathbf{f}: \mathbb{R}^2 \to \mathbb{R}^3$ has how many component scalar functions?

Question 2

A vector-valued function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$ is continuous if and only if each of its component functions is continuous.

Common Mistakes

Confusing the domain dimension $n$ with the output dimension $m$ .
Forgetting that properties like continuity and differentiability are inherited from component functions.
Treating the Jacobian as an $n\times m$ matrix instead of $m\times n$ .