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Unformatted text preview: 3.1. EXTERIOR ALGEBRA 59 â€¢ Also, there holds Î´ i 1 ... i p j 1 ... j p = p ! Î´ i 1 [ j 1 Â·Â·Â· Î´ i p j p ] . â€¢ Thus, the Kronecker symbols Î´ i 1 ... i p j 1 ... j p are the components of the tensors p !Alt( I âŠ— Â·Â·Â· âŠ— I  {z } p ) of type ( p , p ), which are antisymmetric separately in upper indices and the lower indices. â€¢ Thus, the antisymmetrization can also be written as T [ i 1 ... i p ] = 1 p ! Î´ j 1 ... j p i 1 ... i p T j 1 ... j p . â€¢ Notation. â€¢ Obviously, the Kronecker symbols vanish for p > n Î´ i 1 ... i p j 1 ... j p = if p > n . â€¢ The contraction of Kronecker symbols gives Kronecker symbols with lower indices, more precisely, we have the theorem. â€¢ Theorem 3.1.2 For any p , q âˆˆ N , 1 â‰¤ p , q â‰¤ n, there holds Î´ i 1 ... i p l 1 ... l q j 1 ... j p l 1 ... l q = ( n p )! ( n q )! Î´ i 1 ... i p j 1 ... j p . Proof : 1. â€¢ Corollary 3.1.1 For any q âˆˆ N , 1 â‰¤ q â‰¤ n we have Î´ i 1 ... i q i 1 ... i q = n !...
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 Spring '10
 Wong
 Algebra, Geometry, Differential form, Kronecker, Kronecker symbols

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