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sleventh
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Symbolic Linear algebra concept
Evaluate the sum [tex]\Sigma[/tex]k ( [tex]\epsilon[/tex] ijk [tex]\epsilon[/tex] lmk (which contains three terms) by considering the result for all possible combinations of i,j,l,m, that is a) i=j b) i=l c) i=m d) j=l e)j=m f)l=m g) i[tex]\neq[/tex]l or m h) j[tex]\neq[/tex]l or m
show that
[tex]\Sigma[/tex]k ( [tex]\epsilon[/tex] ijk [tex]\epsilon[/tex] lmk = [tex]\delta[/tex] il [tex]\delta[/tex] jm - [tex]\delta[/tex] I am [tex]\delta[/tex] jl
and use this result to prove
A X (B X C)=(A [tex]\bullet[/tex] C)B-(A [tex]\bullet[/tex] B)C
i have been working on this problem for over three hours and have also gone to my professor which was of little help, the work I have done would most likely only serve to confuse. I would be EXTREMELY grateful to anyone who would be able to help me understand this problem. I understand for part a) the permittivity will always equal 0 for the first Levi-Civita symbol making the entire sum zero. For question like be I am lost as to how you could know which values would be +1 or -1 or 0 out of the 81 possible combinations. As for the proofs I am lost on the first and the second I am having a hard time manipulating the concept A X B = [tex]\Sigma[/tex] j, k [tex]\epsilon[/tex] ijk AjBk . Thank you very much to anyone who can shed the slightest of light.
yours truly
sleventh
Homework Statement
Evaluate the sum [tex]\Sigma[/tex]k ( [tex]\epsilon[/tex] ijk [tex]\epsilon[/tex] lmk (which contains three terms) by considering the result for all possible combinations of i,j,l,m, that is a) i=j b) i=l c) i=m d) j=l e)j=m f)l=m g) i[tex]\neq[/tex]l or m h) j[tex]\neq[/tex]l or m
show that
[tex]\Sigma[/tex]k ( [tex]\epsilon[/tex] ijk [tex]\epsilon[/tex] lmk = [tex]\delta[/tex] il [tex]\delta[/tex] jm - [tex]\delta[/tex] I am [tex]\delta[/tex] jl
and use this result to prove
A X (B X C)=(A [tex]\bullet[/tex] C)B-(A [tex]\bullet[/tex] B)C
Homework Equations
The Attempt at a Solution
i have been working on this problem for over three hours and have also gone to my professor which was of little help, the work I have done would most likely only serve to confuse. I would be EXTREMELY grateful to anyone who would be able to help me understand this problem. I understand for part a) the permittivity will always equal 0 for the first Levi-Civita symbol making the entire sum zero. For question like be I am lost as to how you could know which values would be +1 or -1 or 0 out of the 81 possible combinations. As for the proofs I am lost on the first and the second I am having a hard time manipulating the concept A X B = [tex]\Sigma[/tex] j, k [tex]\epsilon[/tex] ijk AjBk . Thank you very much to anyone who can shed the slightest of light.
yours truly
sleventh
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