differential geometry w notes from teacher_Part_30

differential geometry w notes from teacher_Part_30 - 3.1....

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Unformatted text preview: 3.1. EXTERIOR ALGEBRA 59 Also, there holds i 1 ... i p j 1 ... j p = p ! i 1 [ j 1 i p j p ] . Thus, the Kronecker symbols i 1 ... i p j 1 ... j p are the components of the tensors p !Alt( I I | {z } p ) of type ( p , p ), which are anti-symmetric separately in upper indices and the lower indices. Thus, the anti-symmetrization can also be written as T [ i 1 ... i p ] = 1 p ! j 1 ... j p i 1 ... i p T j 1 ... j p . Notation. Obviously, the Kronecker symbols vanish for p > n i 1 ... i p j 1 ... j p = if p > n . The contraction of Kronecker symbols gives Kronecker symbols with lower indices, more precisely, we have the theorem. Theorem 3.1.2 For any p , q N , 1 p , q n, there holds i 1 ... i p l 1 ... l q j 1 ... j p l 1 ... l q = ( n- p )! ( n- q )! i 1 ... i p j 1 ... j p . Proof : 1. Corollary 3.1.1 For any q N , 1 q n we have i 1 ... i q i 1 ... i q = n !...
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differential geometry w notes from teacher_Part_30 - 3.1....

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