LectureNotes13G

LectureNotes13G - Math 6455 Oct 10, 2006 1 Differential...

Info iconThis preview shows pages 1–3. 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: Math 6455 Oct 10, 2006 1 Differential Geometry I Fall 2006, Georgia Tech Lecture Notes 13 Integration on Manifolds, Volume, and Partitions of Unity Suppose that we have an orientable Riemannian manifold ( M,g ) and a function f : M R . How can we define the integral of f on M ? First we answer this question locally, i.e., if ( U, ) is a chart of M (which preserves the orientation of M ), we define Z U fdv g := Z ( U ) f ( - 1 ( x )) q det( g ij ( - 1 ( x ))) dx, where g ij are the coefficients of the metric g in local coordinates ( U, ). Recall that g ij ( p ) := g ( E i ( p ) ,E j ( p )) , where E i ( p ) := d- 1 ( p ) ( e i ) . Now note that if ( V, ) is any other (orientation preserving) local chart of M , and W := U V , then there are two ways to compute R W fdv g , and for these to yield the same answer we need to have Z ( W ) f ( - 1 ( x )) q det( g ij ( - 1 ( x ))) dx = Z ( W ) f ( - 1 ( x )) q det( g ij ( - 1 ( x ))) dx. (1) To check whether the above expression is valid recall that the change variables formula tells that if D R n is an open subset, f : D R is some function, and u : D D is a diffeomorphism, then Z D f ( x ) dx = Z D f ( u ( x ))det( du x ) dx. Now recall that, by the definition of manifolds, - 1 : ( W ) ( W ) is a diffeo- morphism. So, by the change of variables formula, the integral on the left hand side of (1) may be rewritten as Z ( W ) f ( - 1 ( x )) q det( g ij ( - 1 ( x )))det( d ( )- 1 x ) dx. 1 Last revised: November 23, 2009 1 So for equality in (1) to hold we just need to check that q det( g ij ( - 1 ( x ))) = q det( g ij ( - 1 ( x )))det( d ( - 1 ) x ) , for all x ( W ) or, equivalently, q det( g ij ( p )) = q det( g ij ( p ))det( d ( - 1 ) ( p ) ) , (2) for all p W . To see that the above equality holds, let ( a ij ) be the matrix of the linear transformation d ( - 1 ) and note that g ij = g ( d- 1 ( e i ) ,d- 1 ( e j )) = g ( d- 1 d ( - 1 )( e i ) ,d- 1 d (...
View Full Document

Page1 / 5

LectureNotes13G - Math 6455 Oct 10, 2006 1 Differential...

This preview shows document pages 1 - 3. Sign up to view the full document.

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