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Unformatted text preview: p-forms and ( n-p )-forms deﬁned by the Hodge star operator * : Λ p → Λ n-p , that maps any p-form α to a ( n-p )-form * α dual to α deﬁned as follows. • For each p-form α the form * α is the unique ( n-p )-form such that for any p-form β β ∧ * α = h β, α i vol . • In components, this means that ( * α ) i p + 1 ... i n = 1 p ! ε i 1 ... i p i p + 1 ... i n p | g | g i 1 j 1 ··· g i p j p α j 1 ... j p = 1 p ! 1 p | g | g i p + 1 j p + 1 ··· g i n j n ε j 1 ... j p j p + 1 ... j n α j 1 ... j p . • Recall that the Hodge star maps forms to pseudo-forms and vice-versa. • Recall also that for any p-form α , * 2 α = (-1) p ( n-p ) α , meaning that *-1 = (-1) p ( n-p ) * . di ﬀ geom.tex; April 12, 2006; 17:59; p. 188...
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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