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Prove that the permutation symbol (e) defines a Cartesian tensor in R². It is assumed that e; is defined in the same way in every rectangular coordinate system. If the change of coordinates is xax, where (a) (a) = (Spg) and |az| = a₁19229₁2921 = 1 we have to establish the Cartesian tensor law ēj=ersairajs (n=2)

Prove that the permutation symbol (e) defines a Cartesian tensor in R². It is assumed that e; is defined in the same way in every rectangular coordinate system. If the change of coordinates is xax, where (a) (a) = (Spg) and |az| = a₁19229₁2921 = 1 we have to establish the Cartesian tensor law ēj=ersairajs (n=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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Cartesian tensors

Prove that the permutation symbol (e) defines a Cartesian tensor in R². It is assumed that e; is defined in the same way in every rectangular coordinate system. If the change of coordinates is x₁ = a₁x₁, where (a) (a) = (Spg) and |a₁|= 9₁1 922 912921 = 1 we have to establish the Cartesian tensor law ēj=ersair a js Let's examine separately the four possible cases: i=j=1 i=1, j = 2 i=2, j = 1 i=j=2 (n = 2) e9₁a1s = a11912 912911 = 0 =ē11 e,,a1ra28 = a₁1922 912921 = 1 = 12 e,a2ra 18 =a21a12-a22911 = -1 = ²21 ersa₂, a28 = a21922-a22921 = 0 = ²22
Transcribed Image Text:Prove that the permutation symbol (e) defines a Cartesian tensor in R². It is assumed that e; is defined in the same way in every rectangular coordinate system. If the change of coordinates is x₁ = a₁x₁, where (a) (a) = (Spg) and |a₁|= 9₁1 922 912921 = 1 we have to establish the Cartesian tensor law ēj=ersair a js Let's examine separately the four possible cases: i=j=1 i=1, j = 2 i=2, j = 1 i=j=2 (n = 2) e9₁a1s = a11912 912911 = 0 =ē11 e,,a1ra28 = a₁1922 912921 = 1 = 12 e,a2ra 18 =a21a12-a22911 = -1 = ²21 ersa₂, a28 = a21922-a22921 = 0 = ²22
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