differential geometry w notes from teacher_Part_59

# Differential - 5.2 LIE DERIVATIVE OF FORMS AND TENSORS 117 • Recall that a linear map A Λ p → Λ p r is a derivation if r is even and for any

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: 5.2. LIE DERIVATIVE OF FORMS AND TENSORS 117 • Recall that a linear map A : Λ p → Λ p + r is a derivation if r is even and for any α ∈ Λ p and β ∈ Λ q A ( α ∧ β ) = ( A α ) ∧ β + α ∧ A β . and an anti-derivation if r is odd and A ( α ∧ β ) = ( A α ) ∧ β + (- 1) p α ∧ A β . • Examples. • The Lie derivative L X Λ p → Λ p is a derivation. • The exterior derivative d : Λ p → Λ p + 1 and the interior product i X : Λ p → Λ p- 1 are anti-derivations. • Recall the following theorem about exterior derivative Theorem 5.2.7 Let Y 1 , . . . , Y p + 1 be vector fields on a manifold M and α ∈ Λ p be a p-form. Then ( d α )( Y 1 , . . . , Y p + 1 ) = p + 1 X i = 1 (- 1) i + 1 Y i α ( Y 1 , . . . , ˆ Y i , . . . , Y p + 1 ) + X 1 ≤ i < j ≤ p + 1 (- 1) i + j α ([ Y i , Y j ] , . . . , ˆ Y i , . . . , ˆ Y j , . . . , Y p + 1 ) . • In particular, for p = 1 this takes the form Theorem 5.2.8 Let X and Y be vector fields on a manifold M and...
View Full Document

## This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.

### Page1 / 2

Differential - 5.2 LIE DERIVATIVE OF FORMS AND TENSORS 117 • Recall that a linear map A Λ p → Λ p r is a derivation if r is even and for any

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

View Full Document
Ask a homework question - tutors are online