Definition:Definite Integral of Vector-Valued Function

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Let $I \:= \left[{a \,.\,.\, b}\right] \subset \R$ be a closed real interval.

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

$\forall x \in I: \mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k$


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

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

Let $\mathbf g \left({x}\right) := \dfrac \d {\d x} \mathbf f \left({x}\right)$ 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:

$\displaystyle \int_a^b \mathbf g \left({x}\right) \rd x := \mathbf f \left({b}\right) - \mathbf f \left({a}\right)$