Get Region Mathematics essential facts below. View Videos or join the Region Mathematics discussion. Add Region Mathematics to your PopFlock.com topic list for future reference or share this resource on social media.
Various degrees of smoothness of the boundary of the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (Green's theorem, Stokes theorem), properties of Sobolev spaces, and to define measures on the boundary and spaces of traces (generalized functions defined on the boundary). Commonly considered types of domains are domains with continuous boundary, Lipschitz boundary, C1 boundary, and so forth.
According to Hans Hahn, the concept of a domain as an open connected set was introduced by Constantin Carathéodory in his famous book (Carathéodory 1918). Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as a synonym of open set. The rough concept is older. In the 19th and early 20th century, the terms domain and region were often used informally (sometimes interchangeably) without explicit definition.
However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influential monographs on elliptic partial differential equations, Carlo Miranda uses the term "region" to identify an open connected set, and reserves the term "domain" to identify an internally connected,perfect set, each point of which is an accumulation point of interior points, following his former master Mauro Picone: according to this convention, if a set A is a region then its closureA is a domain.
According to Kreyszig,
A region is a set consisting of a domain plus, perhaps, some or all of its boundary points. (The reader is warned that some authors use the term "region" for what we call a domain [following standard terminology], and others make no distinction between the two terms.)
According to Yue Kuen Kwok,
An open connected set is called an open region or domain. ...to an open region we may add none, some, or all its limit points, and simply call the new set a region.
^English: "An open set is connected if it cannot be expressed as the sum of two open sets. An open connected set is called a domain": in this definition, Carathéodory considers obviously non-emptydisjoint sets.
^Hahn (1921, p. 61 footnote 3), commenting the just given definition of open set ("offene Menge"), precisely states:-"Vorher war, für diese Punktmengen die Bezeichnung "Gebiet" in Gebrauch, die wir (§ 5, S. 85) anders verwenden werden." (Free English translation:-"Previously, the term "Gebiet" was occasionally used for such point sets, and it will be used by us in (§ 5, p. 85) with a different meaning."
^Precisely, in the first edition of his monograph, Miranda (1955, p. 1) uses the Italian term "campo", meaning literally "field" in a way similar to its meaning in agriculture: in the second edition of the book, Zane C. Motteler appropriately translates this term as "region".
^An internally connected set is a set whose interior is connected.