Levi-Civita

# Levi-Civita - one place on the previous page Use of the ±...

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Some important Symbols (based on http://icl.pku.edu.cn/yujs/MathWorld/math/k/k137.htm and http://icl.pku.edu.cn/yujs/MathWorld/math/p/p267.htm) Kronecker Delta symbol is deﬁned by deﬁned by δ i j = ˆ e i · ˆ e j , where i, j, k are indices denoting three perpendicular directions in space. The scalar product of two vectors ~ A and ~ B can be written as ~ A · ~ B = ± X i A i ˆ e i ! · ± X j B j ˆ e j ! = X i,j A i B j ˆ e i · ˆ e j = X i,j A i B j δ i,j = X i A i B i A i B i A three-index object sometimes called the Levi-Civita Symbol is deﬁned by ± ij k = ˆ e i · e j × ˆ e k ) = (ˆ e i × ˆ e j ) · ˆ e k It is also known as the permutation symbol. Some of the properties of these symbols are: δ i i = 3 δ ij ± ij k = 0 ± ij k ± ij k 0 = 2 δ kk 0 ± ij k ± ilm = δ j l δ km - δ j m δ k l Interchange of any two indices of the ± symbol reverses its sign. In above mathematical identities, we have implicitly assumed Einstein’s summation conven-

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tion, i.e., repeated indices are understood to be summed over. We already used it
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Unformatted text preview: one place on the previous page. Use of the ± symbol ± ~ A × ~ B ² = X i A i ˆ e i × X j B j ˆ e j = X i, j A i B j (ˆ e i × ˆ e j ) = X i, j A i B j X k [(ˆ e i × ˆ e j ) · ˆ e k ] ˆ e k = X i, j A i B j X k [ ± ij k ] ˆ e k = X k ˆ e k X i, j A i B j [ ± ij k ] | {z } ( ~ A × ~ B ) k ± ~ A × ~ B ² k = X i, j A i B j [ ± ij k ] = ± k ij A i B j ± ~ A × ~ B ² × ~ C = ? h± ~ A × ~ B ² × ~ C i k = ± k ij ± ~ A × ~ B ² i C j = ± k ij ( ± i l m A l B m ) C j = ± ij k ( ± i l m A l B m ) C j = ± ij k ± i l m ( A l B m C j ) = ( δ j l δ k m-δ j m δ k l ) ( A l B m C j ) = ± ~ A · ~ C ² B k-± ~ B · ~ C ² A k ± ~ A × ~ B ² × ~ C = ± ~ A · ~ C ² ~ B-± ~ B · ~ C ² ~ A...
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Levi-Civita - one place on the previous page Use of the ±...

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