differential geometry w notes from teacher_Part_39

# differential geometry w notes from teacher_Part_39 - 77 3.2...

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3.2. ORIENTATION AND THE VOLUME FORM 77 The components of the volume form are vol ( e i 1 , . . . , e i n ) = p | g | ε i 1 ... i n = E i 1 ... i n . 3.2.7 Star Operator and Duality The volume form allows one to deﬁne the duality of p -forms and ( n - p )- vectors. For each p -form A i 1 ... i p one assigns the dual ( n - p )-vector by ˜ A j 1 ... j n - p = 1 p ! E j 1 ... j n - p i 1 ... i p A i 1 ... i p . Similarly, for each p -vector A i 1 ... i p one assigns the dual ( n - p )-form by ˜ A j 1 ... j n - p = 1 p ! E j 1 ... j n - p i 1 ... i p A i 1 ... i p . By lowering and raising the indices of the dual forms we can deﬁne the duality of forms and poly-vectors separately. The Hodge star operator * : Λ p Λ n - p 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 α ∧ * α = ( α, α )vol . In particular,

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differential geometry w notes from teacher_Part_39 - 77 3.2...

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