- #1
Rasalhague
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In Introduction to Vector Analysis, § 1.16 Tensor notation, Davis and Snider introduce index notation and the Einstein summation convention, Kronecker's delta and the Levi-Civita symbol. They present the following equation, on which they base some proofs of vector algebra identities:
[tex]\epsilon_{ikm} \epsilon_{psm} = \delta_{ip} \delta_{ks} - \delta_{is} \delta_{kp}[/tex]
What's puzzling me is: which element on the right expresses the condition that the equation comes to zero if m equals any of the other variables (i,k,p,s), given that m doesn't even appear explicitly on the right?
E.g., from the right-hand side, we can tell that
[tex]\\\mathrm{if} \left(i = p \: \mathrm{and} \: k = s \right) \mathrm{and} \left( i\neq s \: \mathrm{or} \: k\neq p \right)[/tex]
[tex]\mathrm{then} \: \delta_{ip} \delta_{ks} - \delta_{is} \delta_{kp} = 1[/tex]
But what if m = i, say? Then the left-hand side, the product of the two epsilons, would equal 0, wouldn't it? I'm struggling to see how m being equal to one of the other four variables is inconsistent with any of the conditions in the above example for the right-hand side being equal to 1.
I also have a more general question about index notation, about subscripts and superscripts and the terms covariant and contravariant, which I posted in the Linear & Abstract Algebra forum, but maybe it belongs in this forum; I wasn't sure.
https://www.physicsforums.com/showthread.php?t=324814
[tex]\epsilon_{ikm} \epsilon_{psm} = \delta_{ip} \delta_{ks} - \delta_{is} \delta_{kp}[/tex]
What's puzzling me is: which element on the right expresses the condition that the equation comes to zero if m equals any of the other variables (i,k,p,s), given that m doesn't even appear explicitly on the right?
E.g., from the right-hand side, we can tell that
[tex]\\\mathrm{if} \left(i = p \: \mathrm{and} \: k = s \right) \mathrm{and} \left( i\neq s \: \mathrm{or} \: k\neq p \right)[/tex]
[tex]\mathrm{then} \: \delta_{ip} \delta_{ks} - \delta_{is} \delta_{kp} = 1[/tex]
But what if m = i, say? Then the left-hand side, the product of the two epsilons, would equal 0, wouldn't it? I'm struggling to see how m being equal to one of the other four variables is inconsistent with any of the conditions in the above example for the right-hand side being equal to 1.
I also have a more general question about index notation, about subscripts and superscripts and the terms covariant and contravariant, which I posted in the Linear & Abstract Algebra forum, but maybe it belongs in this forum; I wasn't sure.
https://www.physicsforums.com/showthread.php?t=324814