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Unformatted text preview: 3.2. ORIENTATION AND THE VOLUME FORM 75 • Theorem 3.2.3 Let g ij be the components of a Riemannian metric,  g  = det( g ij ) , and ε i 1 ... i n and ε i 1 ... i n be the LeviCivita symbols and E i 1 ... i n and E i 1 ... i n be defined by E i 1 ... i n = p  g  ε i 1 ... i n , E i 1 ... i n = 1 p  g  ε i 1 ... i n Then 1. ε i 1 ... i n represents the components of a pseudonform (that is, a pseudotensor density of type (0 , n ) ) of weight ( 1) . 2. ε i 1 ... i n represents the components of a pseudonvector (that is, a pseudotensor density of type ( n , 0) ) of weight 1 . 3. E i 1 ... i n represents the components of a pseudonform. 4. E i 1 ... i n represents the components of a pseudonvector. Proof : 1. Check the transformation law. 3.2.6 Volume Form • Let { v (1) , . . . , v ( n ) } be an ordered ntuple of vectors. The volume of the par allelepiped spanned by the vectors { v (1) , . . . , v ( n ) } is a real number defined by  vol ( v (1) , . . . , v ( n )...
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 Spring '10
 Wong
 Linear Algebra, Geometry, Differential form, Det, Volume form, Metric tensor, Pseudotensor

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