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Unformatted text preview: 7.5. HARMONIC FORMS 191 • The coderivative is a linear map δ : Λ p → Λ p 1 defined by δ = * 1 d * = ( 1) ( n p + 1)( p 1) * d * . • The exterior derivative and the coderivative satisfy the important conditions d 2 = δ 2 = . • Problem . Show that in local coordinates the coderivative of a pform α is the ( p 1)form δα with components ( δα ) i 1 ... i p 1 = g i 1 j 2 ··· g i p 1 j p 1 p  g  ∂ j p  g  g jk 1 g j 2 k 2 ··· g j p k p α k 1 k 2 ... k p • Now we define the L 2inner product of pforms by ( α, β ) L 2 = Z M α ∧ * β = Z M h α, β i vol , and the L 2norm  α  L 2 = p ( α, α ) L 2 . • This makes the space C ∞ ( Λ p ( M )) of smooth pforms an innerproduct vec tor space. • The completion of C ∞ ( Λ p ( M )) in the L 2norm gives the Hilbert space L 2 ( Λ p ( M )) of squareintegrable pforms. • Let A : H → H be an operator on a Hilbert space H . The adjoint of the operator A with respect to the inner product of the space...
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry, Derivative

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