differential geometry w notes from teacher_Part_41

differential geometry w notes from teacher_Part_41 - 3.3...

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Unformatted text preview: 3.3. EXTERIOR DERIVATIVE 81 • Theorem 3.3.1 The exterior derivative is a linear map d : Λ p → Λ p + 1 . Proof : Show that d α is a form whose value does not depend on the coordi- nate system. • Theorem 3.3.2 Let α ∈ Λ p be a p-form and { v 1 , . . . , v p + 1 } be a collec- tion of ( p + 1) vectors. Then ( d α )( v 1 , . . . , v p + 1 ) = p + 1 X k = 1 (- 1) k- 1 v k ( α ( v 1 , . . . , v k- 1 , v k + 1 , . . . , v p + 1 ))- p + 1 X k = 1 k- 1 X i = 1 (- 1) i + k- 1 α ([ v i , v k ] , v 1 , . . . , v i , v i + 1 , . . . , v k- 1 , v k + 1 , . . . , v p + 1 )) Proof : Calculation. • Remark. This formula can be taken as the intrinsic definition of the exterior derivative. • Theorem 3.3.3 For any p-form d 2 = . Proof : Easy. • Theorem 3.3.4 The exterior derivative d : Λ → Λ is an anti- derivation on the exterior algebra. That is, for any p-form α ∈ Λ p and any q-form β ∈ Λ q there holds d ( α ∧ β ) = ( d α ) ∧ β + (- 1) p α ∧ ( d β ) ....
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differential geometry w notes from teacher_Part_41 - 3.3...

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