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differential geometry w notes from teacher_Part_40

differential geometry w notes from teacher_Part_40 - 79 3.2...

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3.2. ORIENTATION AND THE VOLUME FORM 79 or, in components, α j = E i 1 ... i n - 1 j v i 1 (1) · · · v i n - 1 ( n - 1) . Theorem 3.2.8 Let { v (1) , . . . , v ( n - 1) } be an ordered ( n - 1) -tuple of lin- early independent vectors and { ω (1) , . . . , ω ( n - 1) } be the corresponding 1 -forms. Let α be 1 -form defined by α = * ω (1) ∧ · · · ∧ ω ( n - 1) , and N be the corresponding vector, that is, N i = g ik | g | ε i 1 ... i n - 1 k v i 1 (1) · · · v i n - 1 ( n - 1) Then: 1. The vector N is orthogonal to all vectors { v (1) , . . . , v ( n - 1) } , that is, ( N , v ( j ) ) = g ik N i v k ( j ) = 0 , ( j = 1 , . . . , n - 1) . 2. The n-tuple { v 1 , . . . , v n - 1 , N } forms a positively oriented basis. 3. The volume of the parallelepiped spanned by the vectors { v (1) , . . . , v ( n - 1) , N } is determined by the norm of N vol ( v 1 , . . . , v n - 1 , N ) = ( N , N ) . di ff geom.tex; April 12, 2006; 17:59; p. 81
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80 CHAPTER 3. DIFFERENTIAL FORMS 3.3 Exterior Derivative From now on, if not specified otherwise, we will denote the derivatives by i = x i . The exterior derivative of a 0-form (that is, a function) f is a 1-form d f defined by: for any vector v ( d f )( v
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