Riemann Integral of a Vector-Valued or Matrix-Valued Map

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $p,q\in\mathbb{N}$, let $E\subseteq\mathbb{R}$, let $x:E\to\mathbb{R}^{q}$ and let $A:E\to\mathcal{M}_{p\times q}(\mathbb{R})$. For $k\in[q]$, $x_{k}:E\to\mathbb{R}$ is the component map $s\mapsto x(s)_{k}$, with the components of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii), as in Component Maps and Entry Maps (i); for $i\in[p]$ and $j\in[q]$, $A_{ij}:E\to\mathbb{R}$ is the entry map $s\mapsto A(s)_{ij}$, with the entries of Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii), as in Component Maps and Entry Maps (ii).

(i) Let $u,v\in\mathbb{R}$ with $u<v$ and $[u,v]\subseteq E$. Then $x$ is integrable on $[u,v]$ if $x_{k}$ is integrable on $[u,v]$ for every $k\in[q]$.

(ii) Let $u,v\in E$. The integral $\int_{u}^{v}x$ is defined if the integral $\int_{u}^{v}x_{k}$ is defined for every $k\in[q]$, and then the integral of $x$ from $u$ to $v$ is the point

$$\int_{u}^{v}x=\int_{u}^{v}x(t)\,dt\in\mathbb{R}^{q}\qquad\text{with components}\qquad\Bigl(\int_{u}^{v}x\Bigr)_{k}=\int_{u}^{v}x_{k}\quad(k\in[q]),$$

the numbers $\int_{u}^{v}x_{k}$ being those of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii): the map $[q]\to\mathbb{R}$, $k\mapsto\int_{u}^{v}x_{k}$, is formed by A Predicative Formula for the Oriented Riemann Integral, and Maps Given by Integrals (i) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v), and is a point of $\mathbb{R}^{q}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii), determined by its components by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iii).

(iii) Let $u,v\in\mathbb{R}$ with $u<v$ and $[u,v]\subseteq E$. Then $A$ is integrable on $[u,v]$ if $A_{ij}$ is integrable on $[u,v]$ for all $i\in[p]$ and $j\in[q]$.

(iv) Let $u,v\in E$. The integral $\int_{u}^{v}A$ is defined if the integral $\int_{u}^{v}A_{ij}$ is defined for all $i\in[p]$ and $j\in[q]$, and then the integral of $A$ from $u$ to $v$ is the matrix

$$\int_{u}^{v}A=\int_{u}^{v}A(t)\,dt\in\mathcal{M}_{p\times q}(\mathbb{R})\qquad\text{with entries}\qquad\Bigl(\int_{u}^{v}A\Bigr)_{ij}=\int_{u}^{v}A_{ij}\quad(i\in[p],\ j\in[q]),$$

the numbers $\int_{u}^{v}A_{ij}$ being those of The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii): the map $[p]\times[q]\to\mathbb{R}$, $(i,j)\mapsto\int_{u}^{v}A_{ij}$, is formed by A Predicative Formula for the Oriented Riemann Integral, and Maps Given by Integrals (i) and Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v), and is a matrix by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i), determined by its entries by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii).