Domains and Their Boundaries

Section 10.1 Domains and Their Boundaries

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.

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Definition 10.1. Piecewise smooth domain.

We call

D \subset 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},

where

C_{i}

admits a regular parametrisation

γ_{i} (t),

t \in [a_{i}, b_{i}],

for all

i = 1, \dots, n .

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Given a continuous function

f

defined on the boundary of a piecewise smooth domain

\partial D

we set

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\int_{\partial D} f d s := \int_{C_{1}} f d s + \dots + \int_{C_{n}} f d s .

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.

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Figure 10.2. A piecewise smooth domain with positively oriented boundary.
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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.

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