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 = X i A i e i X j B j e j = X i, j A i B j ( e i e j ) = X i, j A i B j X k [( e i e j ) e k ] e k = X i, j A i B j X k [ ij k ] e k = X k e k X i, j A i B j [ ij k ] | {z } ( ~ A ~ B ) k ~ A ~ B k = X i, j A i B j [ ij k ] = k ij A i B j ~ A ~ B ~ C = ? h ~ A ~ B ~ C i k = k ij ~ A ~ B 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 ) = ( j l k m- j m k l ) ( A l B m C j ) = ~ A ~ C B k- ~ B ~ C A k ~ A ~ B ~ C = ~ A ~ C ~ B- ~ B ~ C ~ A...
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Levi-Civita - one place on the previous page. Use of the...

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