In order to be able to compute an integral over the boundary of a domain we need to assume that it is not too wild. We will work with the following class of domains.
Definition10.1.Piecewise smooth domain.
We call \(D\subset\mathbb R^2\) a piecewise smooth domain if its boundary can be written as a finite union of piecewise smooth curves. More precisely, \(\partial D=C_1\cup C_2\cup\dots\cup C_n\text{,}\) where \(C_i\) admits a regular parametrisation \(\vect\gamma_{i}(t)\text{,}\)\(t\in[a_i,b_i]\text{,}\) for all \(i=1,\dots,n\text{.}\)
Given a continuous function \(f\) defined on the boundary of a piecewise smooth domain \(\partial D\) we set
We orient the boundary of \(D\) in such a way that if we walked along \(\partial D\) in the positive direction then the right arm points out of \(D\) as shown in Figure 10.2.
Figure10.2.A piecewise smooth domain with positively oriented boundary.
Equipped with the above orientation we say that \(\partial D\) (or simply \(D\)) is positively oriented. If we talk about \(\partial D\) we always mean the \textit{positively oriented boundary of \(D\)}! Note that we can also have domains with holes like the one in Figure 10.2. Then the orientation of the inner boundary is opposite to that of the outer boundary.
