In terms of the totally antisymmetric E-symbol (Levi-Civita tensor) with €123 = +1, the vector product can be written as (A x B)i = €ijk Aj Bk, where i, j, k = 1, 2, 3 and summation over repeated indices (here j and k) is implied. i) ii) iii) iv) For general vectors A and B, using (2) prove the following relations: a) A x B=-B x A b) (A x B) A = (A x B) - B = 0. The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j18km - Sjm³ki. (2) Prove the formula (3) €imn€jmn = 2dij. Assuming that (3) is true (and using antisymmetry of the E-symbol), prove the relation A x (B x C) = (AC) B- (AB) C
In terms of the totally antisymmetric E-symbol (Levi-Civita tensor) with €123 = +1, the vector product can be written as (A x B)i = €ijk Aj Bk, where i, j, k = 1, 2, 3 and summation over repeated indices (here j and k) is implied. i) ii) iii) iv) For general vectors A and B, using (2) prove the following relations: a) A x B=-B x A b) (A x B) A = (A x B) - B = 0. The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j18km - Sjm³ki. (2) Prove the formula (3) €imn€jmn = 2dij. Assuming that (3) is true (and using antisymmetry of the E-symbol), prove the relation A x (B x C) = (AC) B- (AB) C
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:In terms of the totally antisymmetric E-symbol (Levi-Civita tensor) with €123 = +1, the vector product
can be written as
(A x B)i = €ijk Aj Bk,
where i, j, k = 1, 2, 3 and summation over repeated indices (here j and k) is implied.
i)
ii)
iii)
iv)
For general vectors A and B, using (2) prove the following relations:
a) A x B=-B x A
b) (A x B) A = (A x B) - B = 0.
The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful
formula
€ijk€ilm = 8j18km - Sjm³ki.
(2)
Prove the formula
(3)
€imn€jmn = 2dij.
Assuming that (3) is true (and using antisymmetry of the E-symbol), prove the relation
A x (B x C) = (AC) B- (AB) C
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