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 \(f_1,\dotsc,f_n\) and let \(S \subseteq \mathbb{R}^m\).

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