differential geometry w notes from teacher_Part_49

differential geometry w notes from teacher_Part_49 - 4.2....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4.2. INTEGRATION OVER MANIFOLDS 97 4.2.2 Integration over Submanifolds A manifold M is compact if every open cover of M has a finite subcover. Thus, very compact manifold has a finite atlas. A subset of R n is compact if it is closed and bounded. Let V be a p-dimensional compact oriented manifold. Let { U } N = 1 be a finite atlas of V . Let each U be positively oriented. Let { } be a partition of unity on V . Let be a p-form over V . The integral of over V is defined by Z V = N X = 1 Z U This integral does not depend on the atlas and the partition of unity. Now, let V be a p-dimensional compact oriented submanifold of an n- dimensional manifold M described by the inclusion map i : V M . Let p M be a p-form on M . The integral of a p-form on M over V M is defined by Z V = Z V i * ....
View Full Document

Page1 / 2

differential geometry w notes from teacher_Part_49 - 4.2....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online