\begin{definition}[Definition:Definite Integral of Vector-Valued Function]
Let $I = \closedint a b \subset \R$ be a closed real interval.
Let $\mathbf f: I \to \R^n$ be a vector-valued function on $I$:
:$\forall x \in I: \map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$
where:
:$f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$
:$\tuple {e_1, e_2, \ldots, e_k}$ denotes the standard ordered basis on $\R^n$.
Let $\mathbf f$ be differentiable on $I$.
Let $\map {\mathbf g} x := \dfrac \d {\d x} \map {\mathbf f} x$ be the derivative of $\mathbf f$ {{WRT|Differentiation}} $x$.
The '''definite integral of $\mathbf g$ {{WRT|Integration}} $x$ from $a$ to $b$''' is defined as:
:$\ds \int_a^b \map {\mathbf g} x \rd x := \map {\mathbf f} b - \map {\mathbf f} a$
\end{definition}