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In terms of the totally antisymmetric e-symbol (Levi-Civita tensor) with €123 = +1, the vector product can be written as (A x B); = ¤ijk Aj Bk, where i, j, k = 1, 2,3 and summation over repeated indices (here j and k) is implied. (a) For general vectors A and B, using (2) prove the following relations: A x B=-B x A (A x B) A = (A x B). B = 0. (b) The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j18km — Sjm kl. (2)

In terms of the totally antisymmetric e-symbol (Levi-Civita tensor) with €123 = +1, the vector product can be written as (A x B); = ¤ijk Aj Bk, where i, j, k = 1, 2,3 and summation over repeated indices (here j and k) is implied. (a) For general vectors A and B, using (2) prove the following relations: A x B=-B x A (A x B) A = (A x B). B = 0. (b) The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j18km — Sjm kl. (2)

Advanced Engineering Mathematics
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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Problem 4 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. (a) For general vectors A and B, using (2) prove the following relations: A x B=-BX A (A x B) A = (A × B) · B = 0. (b) The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j1³km – Sjm³kl. (c) Prove the formula €imn€jmn = 2dij. (d) Assuming that (3) is true (and using antisymmetry of the e-symbol), prove the relation A x (B × C) = (A C) B- (AB) C. (2) (3)
Transcribed Image Text:Problem 4 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. (a) For general vectors A and B, using (2) prove the following relations: A x B=-BX A (A x B) A = (A × B) · B = 0. (b) The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j1³km – Sjm³kl. (c) Prove the formula €imn€jmn = 2dij. (d) Assuming that (3) is true (and using antisymmetry of the e-symbol), prove the relation A x (B × C) = (A C) B- (AB) C. (2) (3)
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