differential geometry w notes from teacher_Part_38

# differential geometry w notes from teacher_Part_38 - 3.2...

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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 Levi-Civita 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 pseudo-n-form (that is, a pseudo-tensor density of type (0 , n ) ) of weight (- 1) . 2. ε i 1 ... i n represents the components of a pseudo-n-vector (that is, a pseudo-tensor density of type ( n , 0) ) of weight 1 . 3. E i 1 ... i n represents the components of a pseudo-n-form. 4. E i 1 ... i n represents the components of a pseudo-n-vector. Proof : 1. Check the transformation law. 3.2.6 Volume Form • Let { v (1) , . . . , v ( n ) } be an ordered n-tuple 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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differential geometry w notes from teacher_Part_38 - 3.2...

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