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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) = (8 pg) and |a₁|= a₁1922912921 = 1 we have to establish the Cartesian tensor law (3.22) ēj=e,sairajs Let's examine separately the four possible cases: (n=2) 1 = ²12 i=j=1 e₁₁ª₁a18 = a11912 912ª911 = 0 = 11 i=1,j-2 e,,a1a28 = a11922-912921 i=2,j=1 e,,azra1s =a21912-a22911 i=j=2 esa₂ra2x=a21922922 921 - 1 = ²21 = - - 0 = €22

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) = (8 pg) and |a₁|= a₁1922912921 = 1 we have to establish the Cartesian tensor law (3.22) ēj=e,sairajs Let's examine separately the four possible cases: (n=2) 1 = ²12 i=j=1 e₁₁ª₁a18 = a11912 912ª911 = 0 = 11 i=1,j-2 e,,a1a28 = a11922-912921 i=2,j=1 e,,azra1s =a21912-a22911 i=j=2 esa₂ra2x=a21922922 921 - 1 = ²21 = - - 0 = €22

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 48E
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(3.22) definition to solve in the other image

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 (3.22) 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 (3.22) 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
A surprising fact is that now covariant and contravariant behaviors are identical. Consequently, all Cartesian tensors will be denoted with subscripts: allowable coordinate changes ;=A₁jXj ○ x₁ = AjjXj cartesian tensor laws T₁ = air Tr Tij = air a js Trs, ... 38 Since an orthogonal transformation converts another rectangular coordinate system into a rectangular one (with the same origin), Cartesian tensors are associated with rectangular (Cartesian) coordinate systems. There are of course more Cartesian tensors than affine ones. Note that JJT = I implies ² = 1₁, or = +1. Objects that obey the laws tensorial (3.22) when the allowable changes in coordinates are such that J = |a₁|= +1 (3.22) They are called proper Cartesian tensors.
Transcribed Image Text:A surprising fact is that now covariant and contravariant behaviors are identical. Consequently, all Cartesian tensors will be denoted with subscripts: allowable coordinate changes ;=A₁jXj ○ x₁ = AjjXj cartesian tensor laws T₁ = air Tr Tij = air a js Trs, ... 38 Since an orthogonal transformation converts another rectangular coordinate system into a rectangular one (with the same origin), Cartesian tensors are associated with rectangular (Cartesian) coordinate systems. There are of course more Cartesian tensors than affine ones. Note that JJT = I implies ² = 1₁, or = +1. Objects that obey the laws tensorial (3.22) when the allowable changes in coordinates are such that J = |a₁|= +1 (3.22) They are called proper Cartesian tensors.
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