# hw9s - H/wk 9(SOLUTIONS Due Wednesday April 4 1 For n ≥ r...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: H/wk 9 (SOLUTIONS). Due Wednesday, April 4. 1. For n ≥ r ≥ 1 define the following function I : Ω r ( R n ) → Ω r- 1 ( R n ) from the space of smooth r-forms on R n to the space of smooth ( r- 1)-forms of R n . For ω = X i 1 <i 2 < ··· <i r ω i 1 ...i r dx i 1 ∧ ··· ∧ dx i r I ( ω ) = X i 1 <i 2 < ··· <i r r X j =1 ( Z 1 (- 1) j +1 t r- 1 ω i 1 ...i r ( tx ) dt ) x i j dx i 1 ∧···∧ d dx i j ∧···∧ dx i r . (a) Prove that ω = I ( dω ) + d ( Iω ) for every r-form ω on R n . (b) Conclude that every closed r-form ω on R n (where r ≥ 1) is exact. That is, show that if dω = 0 then there is some ( r- 1)-form α such that ω = dα . Solution. (a) By linearity of all the formulas involved it suffices to prove that ω = I ( dω ) + d ( Iω ) for all ω of the form ω = f ( x ) dx i 1 ∧ ··· ∧ dx i r where i 1 < i 2 < ··· < i r . We have dω = X k 6 = i 1 ,...,i r ∂f ∂x k dx k ∧ dx i 1 ∧ ··· ∧ dx i r . Then I ( dω ) = X k 6 = i 1 ,...,i r Z 1 t r ∂f ∂x k ( tx ) dt x k dx i 1 ∧ ··· ∧ dx i r + X k 6 = i 1 ,...,i r r X j =1 (- 1) j +2 Z 1 t r ∂f ∂x k ( tx ) dt x i j dx k ∧ dx i 1 ∧ . . . d dx i j ∧ ··· ∧ dx i r . On the other hand, I ( ω ) = r X j =1 (- 1) j +1 Z 1 t r- 1 f ( tx ) dt x i j dx i 1 ∧ . . ....
View Full Document

## This note was uploaded on 04/23/2010 for the course MATH Math 481 taught by Professor Kapovich during the Fall '08 term at University of Illinois at Urbana–Champaign.

### Page1 / 4

hw9s - H/wk 9(SOLUTIONS Due Wednesday April 4 1 For n ≥ r...

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

View Full Document
Ask a homework question - tutors are online