\begin{definition}[Definition:Primitive (Calculus)/Vector-Valued Function]
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 $\mathbf f$ {{WRT|Differentiation}} $x$.
The '''primitive of $\mathbf g$ {{WRT|Integration}} $x$''' 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.
\end{definition}