Integration of Real Vector Functions

Integrals

Definition: Integral of a Real Vector Function

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ be a real vector function with component functions $f_1,\dots ,f_n$ and let $S\subseteq {\mathbb{R}}^m$.

We say that $f$ is integrable on $S$ if $f_1,\dots ,f_n$ are integrable on $S$. In this case, we define the integral of $f$ over $S$ as the real vector

$${\int }_{S}f(\mathbf{x}){\mathrm{d}}^m\mathbf{x}\stackrel{\text{def}}{=}\left[\begin{array}{c}{\int }_S{f}_1(\mathbf{x}){\mathrm{d}}^m\mathbf{x}\\ ⋮\\ {\int }_S{f}_n(\mathbf{x}){\mathrm{d}}^m\mathbf{x}\end{array}\right]$$