Definition:Definite Integral of Vector-Valued Function

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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 $I$:

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


$f_1, f_2, \ldots, f_n$ 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 $\mathbf f$ with respect to $x$.

The definite integral of $\mathbf g$ with respect to $x$ from $a$ to $b$ is defined as:

$\ds \int_a^b \map {\mathbf g} x \rd x := \map {\mathbf f} b - \map {\mathbf f} a$