# Definition:Primitive (Calculus)/Vector-Valued Function

## Definition

Let $U \subset \R$ be an open set in $\R$.

Let $\mathbf f: U \to \R^n$ be a vector-valued function on $U$:

$\forall x \in U: \map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$

where:

$f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$
$\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_k}$ denotes the standard ordered basis on $\R^n$.

Let $\mathbf f$ be differentiable on $U$.

Let $\map {\mathbf g} x := \dfrac \d {\d x} \map {\mathbf f} x$ be the derivative of $\mathbf f$ with respect to $x$.

The primitive of $\mathbf g$ with respect to $x$ is defined as:

$\ds \int \map {\mathbf g} x \rd x := \map {\mathbf f} x + \mathbf c$

where $\mathbf c$ is an arbitrary constant vector.