differential geometry w notes from teacher_Part_43

differential geometry w notes from teacher_Part_43 - * ( )...

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3.5. VECTOR ANALYSIS IN R 3 85 3.5 Vector Analysis in R 3 3.5.1 Vector Algebra in R 3 In the case of three-dimensional Euclidean space the metric in Cartesian coordinates is g i j = δ i j . The bases of p -forms are: 1 , dx , dy , dz , dx dy , dx dz , dy dz , dx dy dz . The star operator acts on this forms by * 1 = dx dy dz , * dx = dy dz , * dy = - dx dz , * dz = dx dy , * ( dx dy ) = dz , * ( dy dz ) = dx , * ( dx dz ) = - dy , * ( dx dy dz ) = 1 . So, any 2-form α = α 12 dx dy + α 13 dx dz + α 23 dy dz is represented by the dual 1-form * α = α 12 dz - α 13 dy + α 23 dx , That is ( * α ) i = 1 2 ε i jk α jk ( * α ) 1 = α 23 , ( * α ) 2 = α 31 , ( * α ) 3 = α 12 , Any 3-form α α = α 123 dx dy dz is represented by the dual 0-form * α = 1 3! ε i jk α i jk = α 123 . di geom.tex; April 12, 2006; 17:59; p. 87
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86 CHAPTER 3. DIFFERENTIAL FORMS Now, let α and β be two 1-forms α = α 1 dx + α 2 dy + α 3 dz , β = β 1 dx + β 2 dy + β 3 dz . Then * β = β 1 dy dz + β 2 dz dx + β 3 dx dz and α β = ( α 1 β 2 - α 2 β 1 ) dx dy + ( α 1 β 3 - α 3 β 1 ) dx dz + ( α 2 β 3 - α 3 β 2 ) dy dz , α ( * β ) = ( α 1 β 1 + α 2 β 2 + α 3 β 3 ) dx dy dz . Therefore,
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Unformatted text preview: * ( ) = ( 1 2- 2 1 ) dz-( 1 3- 3 1 ) dy + ( 2 3- 3 2 ) dx , * [ ( * )] = 1 1 + 2 2 + 3 3 , or * ( ) = , * [ ( * )] = . 3.5.2 Vector Analysis in R 3 Zero-Forms. For a 0-form f we have ( d f ) i = i f , so that d f = grad f . One-Forms. For a 1-form = 1 dx + 2 dy + 3 dz we have ( d ) i j = i j- j i that is, d = ( 1 2- 2 1 ) dx dy + ( 2 3- 3 2 ) dy dz + ( 3 1- 1 3 ) dz dx . di geom.tex; April 12, 2006; 17:59; p. 88...
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differential geometry w notes from teacher_Part_43 - * ( )...

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