nLab area

Contents

Contents

Idea

The volume of a surface.

Definition

Of polygons

Let Polygons\mathrm{Polygons} be the set of all polygons in the Euclidean plane 2\mathbb{R}^2. Then the area is a function A:PolygonsA:\mathrm{Polygons} \to \mathbb{R} such that for all polygons PPolygonsP \in \mathrm{Polygons},

  • AA is invariant under translations:

  • Given a linear transformation LL and a polygon PP, A(LP)=det(L)A(P)A(L P) = \det(L) A(P)

  • Given two vertices pp and qq of PP, …

In terms of Jordan content

Given a large set MM of Jordan-measurable subsets of 2\mathbb{R}^2 bounded by a Jordan curve called shapes, the area of a shape SMS \in M is the Jordan content of SS.

See also

 References

  • Frank Quinn, Proof Projects for Teachers (pdf)

  • Apostol, Tom (1967). Calculus. Vol. I: One-Variable Calculus, with an Introduction to Linear Algebra. pp. 58–59. ISBN 9780471000051.

  • Moise, Edwin (1963). Elementary Geometry from an Advanced Standpoint. Addison-Wesley Pub. Co. (web)

Last revised on June 5, 2022 at 19:16:00. See the history of this page for a list of all contributions to it.