Definition:Vector-Valued Function

Definition

Let $f_1, f_2, \ldots, f_n$ be real functions of $t$.

Let $\mathbb T \subseteq \R, \mathbb Y \subseteq \R^n$ (where usually $n \ge 2$).

Let $\mathbf r$ be a mapping from $\mathbb T \to \mathbb Y$ that maps each $t \in \mathbb T$ to a vector $\tuple {\map {f_1} t, \map {f_2} t, \ldots, \map {f_n} t} \in \mathbb Y$.

Then $\mathbf r$ is said to be a vector-valued function (of the parameter $t$).

If $\mathbb T$ is not explicitly defined, it is taken to be the intersection of all the domains of $f_1 ,f_2, \cdots, f_n$.

Component Function

Each $f_1, f_2, \ldots, f_n$ is a component function of $\mathbf r$.

Also see

• Vector-Valued Function in Terms of Standard Ordered Basis

Sources

• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 12.1$

• For a video presentation of the contents of this page, visit the Khan Academy.