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Unformatted text preview: 3.3. EXTERIOR DERIVATIVE 81 Theorem 3.3.1 The exterior derivative is a linear map d : p p + 1 . Proof : Show that d is a form whose value does not depend on the coordi nate system. Theorem 3.3.2 Let p be a pform and { v 1 , . . . , v p + 1 } be a collec tion of ( p + 1) vectors. Then ( d )( v 1 , . . . , v p + 1 ) = p + 1 X k = 1 ( 1) k 1 v k ( ( 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 ([ 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 pform d 2 = . Proof : Easy. Theorem 3.3.4 The exterior derivative d : is an anti derivation on the exterior algebra. That is, for any pform p and any qform q there holds d ( ) = ( d ) + ( 1) p ( d ) ....
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 Spring '10
 Wong
 Geometry, Derivative

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