2.2 Index Notation for Vector and Tensor Operations

 

 

 

Operations on Cartesian components of vectors and tensors may be expressed very efficiently and clearly using index notation.

 

2.1. Vector and tensor components.

 

Let x be a (three dimensional) vector and let S be a second order tensor.   Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis. Denote the components of x in this basis by ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam iEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4maaqabaaakiaawI cacaGLPaaaaaa@3EEF@ , and denote the components of S by

[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaaeaqabe aacaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaacaaIXaGaaGOmaaqaba GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaSba aSqaaiaaigdacaaIZaaabeaaaOqaaiaadofadaWgaaWcbaGaaGOmai aaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWa aSbaaSqaaiaaikdacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGc baGaam4uamaaBaaaleaacaaIZaGaaGymaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaadofadaWgaaWcbaGaaG4maiaaikdaaeqa aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaBa aaleaacaaIZaGaaG4maaqabaaaaOGaay5waiaaw2faaaaa@81D4@

Using index notation, we would express x and S as

x x i S S ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahIhacqGHHj IUcaWG4bWaaSbaaSqaaiaadMgacaaMc8oabeaakiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaC4uaiabggMi6kaadofadaWgaaWc baGaamyAaiaadQgaaeqaaaaa@57CF@

 

2.2. Conventions and special symbols for index notation

 

 Range Convention: Lower case Latin subscripts (i, j, k…) have the range ( 1,2,3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG ymaiaacYcacaaIYaGaaiilaiaaiodaaiaawIcacaGLPaaaaaa@3B56@ .  The symbol x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaamyAaaqabaaaaa@3850@  denotes three components of a vector x 1 , x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccaGGSaGaaGPaVlaaykW7caWG4bWaaSbaaSqa aiaaikdaaeqaaaaa@3DD2@  and x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaG4maaqabaaaaa@381F@ .  The symbol S ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@391A@  denotes nine components of a second order tensor, S 11 , S 12 , S 13 , S 21 S 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaa WcbaGaaGymaiaaigdaaeqaaOGaaiilaiaaykW7caaMc8UaaGPaVlaa ykW7caWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaacYcacaaMc8 UaaGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaacaaIXaGaaG4maaqa baGccaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaS baaSqaaiaaikdacaaIXaaabeaakiaaykW7caaMc8UaeSOjGSKaaGPa VlaaykW7caWGtbWaaSbaaSqaaiaaiodacaaIZaaabeaaaaa@6038@

 

 Summation convention (Einstein convention): If an index is repeated in a product of vectors or tensors, summation is implied over the repeated index.  Thus

λ= a i b i λ= i=1 3 a i b i λ= a 1 b 1 + a 2 b 2 + a 3 b 3 c i = S ik x k c i = k=1 3 S ik x k { c 1 = S 11 x 1 + S 12 x 2 + S 13 x 3 c 2 = S 21 x 1 + S 22 x 2 + S 23 x 3 c 3 = S 31 x 1 + S 32 x 2 + S 33 x 3 λ= S ij S ij λ= i=1 3 j=1 3 S ij S ij λ= S 11 S 11 + S 12 S 12 ++ S 31 S 31 + S 32 S 32 + S 33 S 33 C ij = A ik B kj C ij = k=1 3 A ik B kj [ C ]=[ A ][ B ] C ij = A ki B kj C ij = k=1 3 A ki B kj [ C ]= [ A ] T [ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeU7aSjabg2da9iaadggada WgaaWcbaGaamyAaaqabaGccaWGIbWaaSbaaSqaaiaadMgaaeqaaOGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq GHHjIUcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH7oaB cqGH9aqpcaaMc8+aaabCaeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaO GaamOyamaaBaaaleaacaWGPbaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaeyyyIORaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7cqaH7oaBcqGH9aqpcaWGHbWaaSbaaSqaaiaa igdaaeqaaOGaamOyamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadg gadaWgaaWcbaGaaGOmaaqabaGccaWGIbWaaSbaaSqaaiaaikdaaeqa aOGaey4kaSIaamyyamaaBaaaleaacaaIZaaabeaakiaadkgadaWgaa WcbaGaaG4maaqabaaabaGaamyAaiabg2da9iaaigdaaeaacaaIZaaa niabggHiLdaakeaacaWGJbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0 Jaam4uamaaBaaaleaacaWGPbGaam4AaaqabaGccaWG4bWaaSbaaSqa aiaadUgaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHHj IUcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadogadaWgaaWcbaGa amyAaaqabaGccqGH9aqpdaaeWbqaaiaadofadaWgaaWcbaGaamyAai aadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaaaeaacaWGRbGa eyypa0JaaGymaaqaaiaaiodaa0GaeyyeIuoakiabggMi6kaaykW7ca aMc8UaaGPaVlaaykW7caaMc8+aaiqaaqaabeqaaiaadogadaWgaaWc baGaaGymaaqabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaaigdacaaIXa aabeaakiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGtbWa aSbaaSqaaiaaigdacaaIYaaabeaakiaadIhadaWgaaWcbaGaaGOmaa qabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaakiaa dIhadaWgaaWcbaGaaG4maaqabaaakeaacaWGJbWaaSbaaSqaaiaaik daaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGc caWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaam4uamaaBaaale aacaaIYaGaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGa ey4kaSIaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccaWG4bWaaS baaSqaaiaaiodaaeqaaaGcbaGaam4yamaaBaaaleaacaaIZaaabeaa kiabg2da9iaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaOGaamiEam aaBaaaleaacaaIXaaabeaakiabgUcaRiaadofadaWgaaWcbaGaaG4m aiaaikdaaeqaaOGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRi aadofadaWgaaWcbaGaaG4maiaaiodaaeqaaOGaamiEamaaBaaaleaa caaIZaaabeaaaaGccaGL7baacaaMc8oabaGaeq4UdWMaeyypa0Jaam 4uamaaBaaaleaacaWGPbGaamOAaaqabaGccaWGtbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaaykW7caaMc8UaaGPaVlabggMi6kaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeU7aSjabg2da9maaqaha baWaaabCaeaacaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaado fadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaGPaVlaaykW7caaMc8Ua aGPaVdWcbaGaamOAaiabg2da9iaaigdaaeaacaaIZaaaniabggHiLd aaleaacaWGPbGaeyypa0JaaGymaaqaaiaaiodaa0GaeyyeIuoakiab ggMi6kaaykW7caaMc8UaaGPaVlabeU7aSjabg2da9iaadofadaWgaa WcbaGaaGymaiaaigdaaeqaaOGaam4uamaaBaaaleaacaaIXaGaaGym aaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaaki aadofadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaey4kaSIaeSOjGSKa ey4kaSIaam4uamaaBaaaleaacaaIZaGaaGymaaqabaGccaWGtbWaaS baaSqaaiaaiodacaaIXaaabeaakiabgUcaRiaadofadaWgaaWcbaGa aG4maiaaikdaaeqaaOGaam4uamaaBaaaleaacaaIZaGaaGOmaaqaba GccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadofa daWgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaam4qamaaBaaaleaaca WGPbGaamOAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaadMgacaWG RbaabeaakiaadkeadaWgaaWcbaGaam4AaiaadQgaaeqaaOGaaGPaVl aaykW7caaMc8UaeyyyIORaaGPaVlaaykW7caaMc8Uaam4qamaaBaaa leaacaWGPbGaamOAaaqabaGccqGH9aqpdaaeWbqaaiaadgeadaWgaa WcbaGaamyAaiaadUgaaeqaaOGaamOqamaaBaaaleaacaWGRbGaamOA aaqabaaabaGaam4Aaiabg2da9iaaigdaaeaacaaIZaaaniabggHiLd GccaaMc8UaaGPaVlaaykW7caaMc8UaeyyyIORaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7daWadaqaaiaadoeaaiaawUfacaGLDbaacqGH9a qpdaWadaqaaiaadgeaaiaawUfacaGLDbaadaWadaqaaiaadkeaaiaa wUfacaGLDbaaaeaacaWGdbWaaSbaaSqaaiaadMgacaWGQbaabeaaki abg2da9iaadgeadaWgaaWcbaGaam4AaiaadMgaaeqaaOGaamOqamaa BaaaleaacaWGRbGaamOAaaqabaGccaaMc8UaaGPaVlaaykW7cqGHHj IUcaaMc8UaaGPaVlaaykW7caWGdbWaaSbaaSqaaiaadMgacaWGQbaa beaakiabg2da9maaqahabaGaamyqamaaBaaaleaacaWGRbGaamyAaa qabaGccaWGcbWaaSbaaSqaaiaadUgacaWGQbaabeaaaeaacaWGRbGa eyypa0JaaGymaaqaaiaaiodaa0GaeyyeIuoakiaaykW7caaMc8UaaG PaVlaaykW7cqGHHjIUcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaa dmaabaGaam4qaaGaay5waiaaw2faaiabg2da9maadmaabaGaamyqaa Gaay5waiaaw2faamaaCaaaleqabaGaamivaaaakmaadmaabaGaamOq aaGaay5waiaaw2faaaaaaa@C185@

 

In the last two equations, [ A ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam yqaaGaay5waiaaw2faaaaa@38F1@ , [ B ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam OqaaGaay5waiaaw2faaaaa@38F2@  and [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam 4qaaGaay5waiaaw2faaaaa@38F3@  denote the ( 3×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG 4maiabgEna0kaaiodaaiaawIcacaGLPaaaaaa@3B53@  component matrices of A, B and C.

 

 The Kronecker Delta:  The symbol δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaWGPbGaamOAaaqabaaaaa@39E7@  is known as the Kronecker delta, and has the properties

δ ij ={ 1i=j 0ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaGabaabaeqabaGaaGym aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaamyAaiabg2da9iaadQgaaeaacaaIWaGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaadMgacqGHGjsUcaWGQbaaaiaawUhaaaaa@647B@

thus

δ 11 = δ 22 = δ 33 =1 δ 12 = δ 21 = δ 13 = δ 31 = δ 32 = δ 32 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaaIXaGaaGymaaqabaGccqGH9aqpcqaH0oazdaWgaaWcbaGa aGOmaiaaikdaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaaiodaca aIZaaabeaakiabg2da9iaaigdacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq aH0oazdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaeqiTdq2a aSbaaSqaaiaaikdacaaIXaaabeaakiabg2da9iabes7aKnaaBaaale aacaaIXaGaaG4maaqabaGccqGH9aqpcqaH0oazdaWgaaWcbaGaaG4m aiaaigdaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaaiodacaaIYa aabeaakiabg2da9iabes7aKnaaBaaaleaacaaIZaGaaGOmaaqabaGc cqGH9aqpcaaIWaaaaa@7153@

You can also think of δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaWGPbGaamOAaaqabaaaaa@39E7@  as the components of the identity tensor, or a ( 3×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG 4maiabgEna0kaaiodaaiaawIcacaGLPaaaaaa@3B53@  identity matrix.  Observe the following useful results

δ ij = δ ji δ kk =3 a i = δ ik a k A ij = δ ik A kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqiTdq 2aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iabes7aKnaaBaaa leaacaWGQbGaamyAaaqabaaakeaacqaH0oazdaWgaaWcbaGaam4Aai aadUgaaeqaaOGaeyypa0JaaG4maaqaaiaadggadaWgaaWcbaGaamyA aaqabaGccqGH9aqpcqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaO GaamyyamaaBaaaleaacaWGRbaabeaaaOqaaiaadgeadaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgaca WGRbaabeaakiaadgeadaWgaaWcbaGaam4AaiaadQgaaeqaaaaaaa@5774@

 

 The Permutation Symbol: The symbol ijk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIGiopaaBa aaleaacaWGPbGaamOAaiaadUgaaeqaaaaa@3AB6@  has properties

ijk ={ 1i,j,k=1,2,3;2,3,1or3,1,2 1i,j,k=3,2,1;2,1,3or 1,3,2 0otherwise MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIGiopaaBa aaleaacaWGPbGaamOAaiaadUgaaeqaaOGaeyypa0Zaaiqaaqaabeqa aiaaigdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caWGPbGaaiilaiaadQgacaGGSaGaam4Aaiabg2da 9iaaigdacaGGSaGaaGOmaiaacYcacaaIZaGaai4oaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaikdacaGGSaGaaG4maiaacYca caaIXaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caqGVbGaaeOCai aaykW7caaMc8UaaGPaVlaaykW7caaIZaGaaiilaiaaigdacaGGSaGa aGOmaaqaaiabgkHiTiaaigdacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaadMgacaGGSaGaamOAaiaacYcacaWGRbGaeyypa0JaaGPaVlaa iodacaGGSaGaaGOmaiaacYcacaaIXaGaai4oaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaikdacaGGSaGaaGymaiaacYcacaaI ZaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaae4Baiaabk hacaqGGaGaaGPaVlaaykW7caaMc8UaaeymaiaabYcacaqGZaGaaeil aiaabkdaaeaacaqGWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaab+gacaqG0bGaaeiAaiaabwgacaqGYbGaae4Daiaa bMgacaqGZbGaaeyzaaaacaGL7baaaaa@BE65@

thus

123 = 231 = 312 =1 321 = 213 = 132 =1 111 = 112 = 113 = 121 = 122 = 131 = 133 =0 211 = 212 = 221 = 222 = 223 = 232 = 233 =0 311 = 313 = 322 = 323 = 321 = 332 = 333 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeyicI4 8aaSbaaSqaaiaaigdacaaIYaGaaG4maaqabaGccqGH9aqpcqGHiiIZ daWgaaWcbaGaaGOmaiaaiodacaaIXaaabeaakiabg2da9iabgIGiop aaBaaaleaacaaIZaGaaGymaiaaikdaaeqaaOGaeyypa0JaaGymaaqa aiabgIGiopaaBaaaleaacaaIZaGaaGOmaiaaigdaaeqaaOGaeyypa0 JaeyicI48aaSbaaSqaaiaaikdacaaIXaGaaG4maaqabaGccqGH9aqp cqGHiiIZdaWgaaWcbaGaaGymaiaaiodacaaIYaaabeaakiabg2da9i abgkHiTiaaigdaaeaacqGHiiIZdaWgaaWcbaGaaGymaiaaigdacaaI Xaaabeaakiabg2da9iabgIGiopaaBaaaleaacaaIXaGaaGymaiaaik daaeqaaOGaeyypa0JaeyicI48aaSbaaSqaaiaaigdacaaIXaGaaG4m aaqabaGccqGH9aqpcqGHiiIZdaWgaaWcbaGaaGymaiaaikdacaaIXa aabeaakiabg2da9iabgIGiopaaBaaaleaacaaIXaGaaGOmaiaaikda aeqaaOGaeyypa0JaeyicI48aaSbaaSqaaiaaigdacaaIZaGaaGymaa qabaGccqGH9aqpcqGHiiIZdaWgaaWcbaGaaGymaiaaiodacaaIZaaa beaakiabg2da9iaaicdaaeaacqGHiiIZdaWgaaWcbaGaaGOmaiaaig dacaaIXaaabeaakiabg2da9iabgIGiopaaBaaaleaacaaIYaGaaGym aiaaikdaaeqaaOGaeyypa0JaeyicI48aaSbaaSqaaiaaikdacaaIYa GaaGymaaqabaGccqGH9aqpcqGHiiIZdaWgaaWcbaGaaGOmaiaaikda caaIYaaabeaakiabg2da9iabgIGiopaaBaaaleaacaaIYaGaaGOmai aaiodaaeqaaOGaeyypa0JaeyicI48aaSbaaSqaaiaaikdacaaIZaGa aGOmaaqabaGccqGH9aqpcqGHiiIZdaWgaaWcbaGaaGOmaiaaiodaca aIZaaabeaakiabg2da9iaaicdaaeaacqGHiiIZdaWgaaWcbaGaaG4m aiaaigdacaaIXaaabeaakiabg2da9iabgIGiopaaBaaaleaacaaIZa GaaGymaiaaiodaaeqaaOGaeyypa0JaeyicI48aaSbaaSqaaiaaioda caaIYaGaaGOmaaqabaGccqGH9aqpcqGHiiIZdaWgaaWcbaGaaG4mai aaikdacaaIZaaabeaakiabg2da9iabgIGiopaaBaaaleaacaaIZaGa aGOmaiaaigdaaeqaaOGaeyypa0JaeyicI48aaSbaaSqaaiaaiodaca aIZaGaaGOmaaqabaGccqGH9aqpcqGHiiIZdaWgaaWcbaGaaG4maiaa iodacaaIZaaabeaakiabg2da9iaaicdaaaaa@C08F@

Note that

ijk = kij = jki = jik = kji = kji kki =0 ijk imn = δ jm δ kn δ jn δ mk ijk lmn = δ il ( δ jm δ kn δ jn δ km ) δ im ( δ jl δ kn δ jn δ kl )+ δ in ( δ jl δ km δ jm δ kl ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeyicI48aaSbaaSqaaiaadMgaca WGQbGaam4AaaqabaGccqGH9aqpcqGHiiIZdaWgaaWcbaGaam4Aaiaa dMgacaWGQbaabeaakiabg2da9iabgIGiopaaBaaaleaacaWGQbGaam 4AaiaadMgaaeqaaOGaeyypa0JaeyOeI0IaeyicI48aaSbaaSqaaiaa dQgacaWGPbGaam4AaaqabaGccqGH9aqpcqGHsislcqGHiiIZdaWgaa WcbaGaam4AaiaadQgacaWGPbaabeaakiabg2da9iabgkHiTiabgIGi opaaBaaaleaacaWGRbGaamOAaiaadMgaaeqaaaGcbaGaeyicI48aaS baaSqaaiaadUgacaWGRbGaamyAaaqabaGccqGH9aqpcaaIWaaabaGa eyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccqGHiiIZda WgaaWcbaGaamyAaiaad2gacaWGUbaabeaakiabg2da9iabes7aKnaa BaaaleaacaWGQbGaamyBaaqabaGccqaH0oazdaWgaaWcbaGaam4Aai aad6gaaeqaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadQgacaWGUbaa beaakiabes7aKnaaBaaaleaacaWGTbGaam4AaaqabaaakeaacqGHii IZdaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaakiabgIGiopaaBaaa leaacaWGSbGaamyBaiaad6gaaeqaaOGaeyypa0JaeqiTdq2aaSbaaS qaaiaadMgacaWGSbaabeaakmaabmaabaGaeqiTdq2aaSbaaSqaaiaa dQgacaWGTbaabeaakiabes7aKnaaBaaaleaacaWGRbGaamOBaaqaba GccqGHsislcqaH0oazdaWgaaWcbaGaamOAaiaad6gaaeqaaOGaeqiT dq2aaSbaaSqaaiaadUgacaWGTbaabeaaaOGaayjkaiaawMcaaiabgk HiTiabes7aKnaaBaaaleaacaWGPbGaamyBaaqabaGcdaqadaqaaiab es7aKnaaBaaaleaacaWGQbGaamiBaaqabaGccqaH0oazdaWgaaWcba Gaam4Aaiaad6gaaeqaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadQga caWGUbaabeaakiabes7aKnaaBaaaleaacaWGRbGaamiBaaqabaaaki aawIcacaGLPaaacqGHRaWkcqaH0oazdaWgaaWcbaGaamyAaiaad6ga aeqaaOWaaeWaaeaacqaH0oazdaWgaaWcbaGaamOAaiaadYgaaeqaaO GaeqiTdq2aaSbaaSqaaiaadUgacaWGTbaabeaakiabgkHiTiabes7a KnaaBaaaleaacaWGQbGaamyBaaqabaGccqaH0oazdaWgaaWcbaGaam 4AaiaadYgaaeqaaaGccaGLOaGaayzkaaaaaaa@BF37@

 

2.3. Rules of index notation

 

1. The same index (subscript) may not appear more than twice in a product of two (or more) vectors or tensors.  Thus

A ik x k , A ik B kj , A ij B ik C nk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaamyAaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa kiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaamyqamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGcbWaaSbaaSqa aiaadUgacaWGQbaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGbbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaO Gaam4qamaaBaaaleaacaWGUbGaam4Aaaqabaaaaa@6387@

are valid, but

A kk x k , A ik B kk , A ij B ik C ik MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaam4AaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa kiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaamyqamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGcbWaaSbaaSqa aiaadUgacaWGRbaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGbbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaO Gaam4qamaaBaaaleaacaWGPbGaam4Aaaqabaaaaa@6385@

are meaningless

 

 

 

 

2. Free indices on each term of an equation must agree.  Thus

x i = u i + c i x=u+c a i = A ki B kj x j + C ik u k a= A T Bx+Cu MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEam aaBaaaleaacaWGPbaabeaakiabg2da9iaadwhadaWgaaWcbaGaamyA aaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eyyyIORaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aahIhacqGH9aqpcaWH1bGaey4kaSIaaC4yaaqaaiaadggadaWgaaWc baGaamyAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaadUgacaWGPb aabeaakiaadkeadaWgaaWcbaGaam4AaiaadQgaaeqaaOGaamiEamaa BaaaleaacaWGQbaabeaakiabgUcaRiaadoeadaWgaaWcbaGaamyAai aadUgaaeqaaOGaamyDamaaBaaaleaacaWGRbaabeaakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaeyyyIORaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaCyyaiabg2da9iaahgeadaahaaWcbeqaaiaadsfaaaGccaWHcb GaaGPaVlaahIhacqGHRaWkcaWHdbGaaCyDaaaaaa@9456@

are valid, but

x i = A ij x j = A ik u k x i = A ik u k + c j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEam aaBaaaleaacaWGPbaabeaakiabg2da9iaadgeadaWgaaWcbaGaamyA aiaadQgaaeqaaaGcbaGaamiEamaaBaaaleaacaWGQbaabeaakiabg2 da9iaadgeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamyDamaaBaaa leaacaWGRbaabeaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWGbbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadwhadaWg aaWcbaGaam4AaaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaadQgaae qaaaaaaa@4F69@

are meaningless.

 

3.  Free and dummy indices may be changed without altering the meaning of an expression, provided that rules 1 and 2 are not violated. Thus

x i = A ik x k x j = A jk x k x j = A ji x i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaamyAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaadMgacaWG RbaabeaakiaadIhadaWgaaWcbaGaam4AaaqabaGccqGHuhY2caWG4b WaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamyqamaaBaaaleaacaWG QbGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyi1HS TaamiEamaaBaaaleaacaWGQbaabeaakiabg2da9iaadgeadaWgaaWc baGaamOAaiaadMgaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaaaa a@5353@

 

2.4. Vector operations expressed using index notation

 

 Addition.   c=a+b c i = a i + b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogacqGH9a qpcaWHHbGaey4kaSIaaCOyaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaadogadaWgaaWcbaGaamyAaaqabaGc cqGH9aqpcaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamOyam aaBaaaleaacaWGPbaabeaaaaa@5BCF@

 

 Dot Product  λ=abλ= a i b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSjabg2 da9iaahggacqGHflY1caWHIbGaaGPaVlaaykW7caaMc8UaaGPaVlab ggMi6kaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4UdWMaeyypa0 JaamyyamaaBaaaleaacaWGPbaabeaakiaadkgadaWgaaWcbaGaamyA aaqabaaaaa@5383@

 

 Vector Product c=a×b c i = ijk a j b k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogacqGH9a qpcaWHHbGaey41aqRaaCOyaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyyyIORaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG JbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGPaVlabgIGiopaaBa aaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWG QbaabeaakiaadkgadaWgaaWcbaGaam4Aaaqabaaaaa@6863@

 

 Dyadic Product   S=ab S ij = a i b j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahofacqGH9a qpcaWHHbGaey4LIqSaaCOyaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaSbaaSqaaiaadMga caWGQbaabeaakiabg2da9iaadggadaWgaaWcbaGaamyAaaqabaGcca WGIbWaaSbaaSqaaiaadQgaaeqaaaaa@5E6F@

 

 Change of Basis.  Let a be a vector. Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis, and denote the components of a in this basis by a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamyAaaqabaaaaa@3839@ .  Let { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yBamaaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F82@  be a second basis, and denote the components of a in this basis by α i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqydaWgaaWcbaGaamyAaaqaba aaaa@357E@ .  Then, define

Q ij = m i e j =cosθ( m i , e j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaCyBamaaBaaaleaacaWG PbaabeaakiabgwSixlaahwgadaWgaaWcbaGaamOAaaqabaGccqGH9a qpciGGJbGaai4BaiaacohacqaH4oqCcaGGOaGaaCyBamaaBaaaleaa caWGPbaabeaakiaacYcacaWHLbWaaSbaaSqaaiaadQgaaeqaaOGaai ykaaaa@4C64@

where θ( m i , e j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaacI cacaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaahwgadaWgaaWc baGaamOAaaqabaGccaGGPaaaaa@3E25@  denotes the angle between the unit vectors m i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa WcbaGaamyAaaqabaaaaa@3849@   and e j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaamOAaaqabaaaaa@3842@ .  Then

α i = Q ij a j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHnaaBa aaleaacaWGPbaabeaakiabg2da9iaadgfadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaamyyamaaBaaaleaacaWGQbaabeaaaaa@3EEC@

 

2.5. Tensor operations expressed using index notation.

 

 Addition.   C=A+B C ij = A ij + B ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbGaeyypa0JaaCyqaiabgUcaRi aahkeacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7cqGHHjIUcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWGdbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaa dgeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaamOqamaaBa aaleaacaWGPbGaamOAaaqabaaaaa@5A68@

 

 Transpose  A= B T A ij = B ji MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHbbGaeyypa0JaaCOqamaaCaaale qabaGaamivaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyyyIORaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadgeadaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyypa0JaamOqamaaBaaaleaacaWGQbGaam yAaaqabaaaaa@5AAE@

 

 Scalar Products λ=A:Bλ= A ij B ij λ=ABλ= A ji B ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeU7aSjabg2da9iaahgeaca GG6aGaaCOqaiaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaeq4UdWMaeyypa0JaamyqamaaBaaaleaaca WGPbGaamOAaaqabaGccaWGcbWaaSbaaSqaaiaadMgacaWGQbaabeaa aOqaaiabeU7aSjabg2da9iaahgeacqGHflY1cqGHflY1caaMc8UaaC OqaiaaykW7caaMc8UaaGPaVlaaykW7cqGHHjIUcqaH7oaBcqGH9aqp caWGbbWaaSbaaSqaaiaadQgacaWGPbaabeaakiaadkeadaWgaaWcba GaamyAaiaadQgaaeqaaaaaaa@692D@

 

 Product of a tensor and a vector c=Ab c i = A ij b j c= A T b c i = A ji b j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahogacqGH9aqpcaWHbbGaaC OyaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlabggMi6kaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGJbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Jaamyq amaaBaaaleaacaWGPbGaamOAaaqabaGccaWGIbWaaSbaaSqaaiaadQ gaaeqaaaGcbaGaaC4yaiabg2da9iaahgeadaahaaWcbeqaaiaadsfa aaGccaWHIbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabggMi6kaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa dogadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaai aadQgacaWGPbaabeaakiaadkgadaWgaaWcbaGaamOAaaqabaaaaaa@7B5F@

 

 Product of two tensors  C=AB C ij = A ik B kj C= A T B C ij = A ki B kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahoeacqGH9aqpcaWHbbGaaC OqaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caWGdbWaaSbaaSqaaiaadMga caWGQbaabeaakiabg2da9iaadgeadaWgaaWcbaGaamyAaiaadUgaae qaaOGaamOqamaaBaaaleaacaWGRbGaamOAaaqabaaakeaacaWHdbGa eyypa0JaaCyqamaaCaaaleqabaGaamivaaaakiaahkeacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH HjIUcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam 4qamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGbbWaaSba aSqaaiaadUgacaWGPbaabeaakiaadkeadaWgaaWcbaGaam4AaiaadQ gaaeqaaaaaaa@8761@

* Determinant λ=detAλ= 1 6 ijk lmn A li A mj A nk = ijk A i1 A j2 A k3 lmn λ= ijk A li A mj A nk = ijk A il A jm A kn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeU7aSjabg2da9iGacsgaca GGLbGaaiiDaiaahgeacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqGHHjIUcaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlabeU7aSjabg2da9iaaykW7caaMc8+aaSaaaeaacaaI XaaabaGaaGOnaaaacqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRb aabeaakiabgIGiopaaBaaaleaacaWGSbGaamyBaiaad6gaaeqaaOGa amyqamaaBaaaleaacaWGSbGaamyAaaqabaGccaWGbbWaaSbaaSqaai aad2gacaWGQbaabeaakiaadgeadaWgaaWcbaGaamOBaiaadUgaaeqa aOGaaGPaVlabg2da9iabgIGiopaaBaaaleaacaWGPbGaamOAaiaadU gaaeqaaOGaamyqamaaBaaaleaacaWGPbGaaGymaaqabaGccaWGbbWa aSbaaSqaaiaadQgacaaIYaaabeaakiaadgeadaWgaaWcbaGaam4Aai aaiodaaeqaaaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlabgsDiBlaaykW7caaMc8UaaGPaVlabgIGiopaaBaaaleaa caWGSbGaamyBaiaad6gaaeqaaOGaeq4UdWMaaGPaVlaaykW7cqGH9a qpcaaMc8UaaGPaVlaaykW7cqGHiiIZdaWgaaWcbaGaamyAaiaadQga caWGRbaabeaakiaadgeadaWgaaWcbaGaamiBaiaadMgaaeqaaOGaam yqamaaBaaaleaacaWGTbGaamOAaaqabaGccaWGbbWaaSbaaSqaaiaa d6gacaWGRbaabeaakiabg2da9iabgIGiopaaBaaaleaacaWGPbGaam OAaiaadUgaaeqaaOGaamyqamaaBaaaleaacaWGPbGaamiBaaqabaGc caWGbbWaaSbaaSqaaiaadQgacaWGTbaabeaakiaadgeadaWgaaWcba Gaam4Aaiaad6gaaeqaaaaaaa@D99F@

* Inverse S ji 1 = 1 2det(S) ipq jkl S pk S ql MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaa0baaS qaaiaadQgacaWGPbaabaGaeyOeI0IaaGymaaaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaciGGKbGaaiyzaiaacshacaGGOaGaaC4uai aacMcaaaGaeyicI48aaSbaaSqaaiaadMgacaWGWbGaamyCaaqabaGc cqGHiiIZdaWgaaWcbaGaamOAaiaadUgacaWGSbaabeaakiaadofada WgaaWcbaGaamiCaiaadUgaaeqaaOGaam4uamaaBaaaleaacaWGXbGa amiBaaqabaaaaa@5092@

 Change of Basis.  Let A be a second order tensor. Let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU hacaGL9baaaaa@3F6A@  be a Cartesian basis, and denote the components of A in this basis by A ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@3594@ .  Let { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0E@  be a second basis, and denote the components of A in this basis by Λ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfU5amnaaBa aaleaacaWGPbGaamOAaaqabaaaaa@39B7@ .  Then, define

Q ij = m i e j =cosθ( m i , e j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaCyBamaaBaaaleaacaWG PbaabeaakiabgwSixlaahwgadaWgaaWcbaGaamOAaaqabaGccqGH9a qpciGGJbGaai4BaiaacohacqaH4oqCcaGGOaGaaCyBamaaBaaaleaa caWGPbaabeaakiaacYcacaWHLbWaaSbaaSqaaiaadQgaaeqaaOGaai ykaaaa@4C64@

where θ( m i , e j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaacI cacaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaahwgadaWgaaWc baGaamOAaaqabaGccaGGPaaaaa@3E25@  denotes the angle between the unit vectors m i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa WcbaGaamyAaaqabaaaaa@3849@   and e j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaamOAaaqabaaaaa@3842@ .  Then

Λ ij = Q ik A km Q jm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfU5amnaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGrbWaaSbaaSqaaiaa dMgacaWGRbaabeaakiaadgeadaWgaaWcbaGaam4Aaiaad2gaaeqaaO GaamyuamaaBaaaleaacaWGQbGaamyBaaqabaaaaa@4372@

 

 

2.6. Calculus using index notation

 

The derivative x i / x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaa aa@3A7E@  can be deduced by noting that x i / x j =1i=j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc cqGH9aqpcaaIXaGaaGPaVlaaykW7caaMc8UaaGPaVlaadMgacqGH9a qpcaWGQbaaaa@4558@  and  x i / x j =0ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc cqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaadMgacqGHGj sUcaWGQbaaaa@4618@ .  Therefore

                                                                  x i x j = δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadIhadaWgaa WcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaaaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqaba aaaa@3E98@

The same argument can be used for higher order tensors

                                                               A ij A kl = δ ik δ jl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadgeadaWgaa WcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaamyqamaaBaaaleaa caWGRbGaamiBaaqabaaaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadM gacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGQbGaamiBaaqabaaa aa@43C7@

 

 

2.7. Examples of algebraic manipulations using index notation

 

1. Let a, b, c, d be vectors.  Prove that

( a×b )( c×d )=( ac )( bd )( bc )( ad ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahggacqGHxdaTcaWHIb aacaGLOaGaayzkaaGaeyyXIC9aaeWaaeaacaWHJbGaey41aqRaaCiz aaGaayjkaiaawMcaaiabg2da9maabmaabaGaaCyyaiabgwSixlaaho gaaiaawIcacaGLPaaadaqadaqaaiaahkgacqGHflY1caWHKbaacaGL OaGaayzkaaGaeyOeI0YaaeWaaeaacaWHIbGaeyyXICTaaC4yaaGaay jkaiaawMcaamaabmaabaGaaCyyaiabgwSixlaahsgaaiaawIcacaGL Paaaaaa@5898@

 

Express the left hand side of the equation using index notation (check the rules for cross products and dot products of vectors to see how this is done)

( a×b )( c×d ) ijk a j b k imn c m d n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahggacqGHxdaTcaWHIb aacaGLOaGaayzkaaGaeyyXIC9aaeWaaeaacaWHJbGaey41aqRaaCiz aaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7cqGHHjIUca aMc8UaaGPaVlaaykW7caaMc8UaeyicI48aaSbaaSqaaiaadMgacaWG QbGaam4AaaqabaGccaWGHbWaaSbaaSqaaiaadQgaaeqaaOGaamOyam aaBaaaleaacaWGRbaabeaakiabgIGiopaaBaaaleaacaWGPbGaamyB aiaad6gaaeqaaOGaam4yamaaBaaaleaacaWGTbaabeaakiaadsgada WgaaWcbaGaamOBaaqabaaaaa@5F62@

Recall the identity

ijk imn = δ jm δ kn δ jn δ mk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQ gacaWGRbaabeaakiabgIGiopaaBaaaleaacaWGPbGaamyBaiaad6ga aeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadQgacaWGTbaabeaaki abes7aKnaaBaaaleaacaWGRbGaamOBaaqabaGccqGHsislcqaH0oaz daWgaaWcbaGaamOAaiaad6gaaeqaaOGaeqiTdq2aaSbaaSqaaiaad2 gacaWGRbaabeaaaaa@4CB6@

so

ijk a j b k imn c m d n =( δ jm δ kn δ jn δ mk ) a j b k c m d n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlabgIGiopaaBaaale aacaWGPbGaamOAaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWGQbaa beaakiaadkgadaWgaaWcbaGaam4AaaqabaGccqGHiiIZdaWgaaWcba GaamyAaiaad2gacaWGUbaabeaakiaadogadaWgaaWcbaGaamyBaaqa baGccaWGKbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0ZaaeWaaeaacq aH0oazdaWgaaWcbaGaamOAaiaad2gaaeqaaOGaeqiTdq2aaSbaaSqa aiaadUgacaWGUbaabeaakiabgkHiTiabes7aKnaaBaaaleaacaWGQb GaamOBaaqabaGccqaH0oazdaWgaaWcbaGaamyBaiaadUgaaeqaaaGc caGLOaGaayzkaaGaamyyamaaBaaaleaacaWGQbaabeaakiaadkgada WgaaWcbaGaam4AaaqabaGccaWGJbWaaSbaaSqaaiaad2gaaeqaaOGa amizamaaBaaaleaacaWGUbaabeaaaaa@61C9@

Multiply out, and note that

δ jm a j = a m δ kn b k = b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazdaWgaaWcbaGaamOAaiaad2 gaaeqaaOGaamyyamaaBaaaleaacaWGQbaabeaakiabg2da9iaadgga daWgaaWcbaGaamyBaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH 0oazdaWgaaWcbaGaam4Aaiaad6gaaeqaaOGaamOyamaaBaaaleaaca WGRbaabeaakiabg2da9iaadkgadaWgaaWcbaGaamOBaaqabaaaaa@56FB@

(multiplying by a Kronecker delta has the effect of switching indices…) so

( δ jm δ kn δ jn δ mk ) a j b k c m d n = a m b n c m d n a n b m c m d n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlaaykW7daqadaqaai abes7aKnaaBaaaleaacaWGQbGaamyBaaqabaGccqaH0oazdaWgaaWc baGaam4Aaiaad6gaaeqaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadQ gacaWGUbaabeaakiabes7aKnaaBaaaleaacaWGTbGaam4Aaaqabaaa kiaawIcacaGLPaaacaWGHbWaaSbaaSqaaiaadQgaaeqaaOGaamOyam aaBaaaleaacaWGRbaabeaakiaadogadaWgaaWcbaGaamyBaaqabaGc caWGKbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamyyamaaBaaale aacaWGTbaabeaakiaadkgadaWgaaWcbaGaamOBaaqabaGccaWGJbWa aSbaaSqaaiaad2gaaeqaaOGaamizamaaBaaaleaacaWGUbaabeaaki abgkHiTiaadggadaWgaaWcbaGaamOBaaqabaGccaWGIbWaaSbaaSqa aiaad2gaaeqaaOGaam4yamaaBaaaleaacaWGTbaabeaakiaadsgada WgaaWcbaGaamOBaaqabaaaaa@6373@

Finally, note that

a m c m ac MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaad2gaaeqaaO Gaam4yamaaBaaaleaacaWGTbaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7cqGHHjIUcaWHHbGaeyyXICTaaC4yaaaa@42F8@

and similarly for other products with the same index, so that

a m b n c m d n a n b m c m d n = a m c m b n d n b m c m a n d n ( ac )( bd )( bc )( ad ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaad2gaaeqaaO GaamOyamaaBaaaleaacaWGUbaabeaakiaadogadaWgaaWcbaGaamyB aaqabaGccaWGKbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0Iaamyyam aaBaaaleaacaWGUbaabeaakiaadkgadaWgaaWcbaGaamyBaaqabaGc caWGJbWaaSbaaSqaaiaad2gaaeqaaOGaamizamaaBaaaleaacaWGUb aabeaakiabg2da9iaadggadaWgaaWcbaGaamyBaaqabaGccaWGJbWa aSbaaSqaaiaad2gaaeqaaOGaamOyamaaBaaaleaacaWGUbaabeaaki aadsgadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGIbWaaSbaaSqa aiaad2gaaeqaaOGaam4yamaaBaaaleaacaWGTbaabeaakiaadggada WgaaWcbaGaamOBaaqabaGccaWGKbWaaSbaaSqaaiaad6gaaeqaaOGa eyyyIO7aaeWaaeaacaWHHbGaeyyXICTaaC4yaaGaayjkaiaawMcaam aabmaabaGaaCOyaiabgwSixlaahsgaaiaawIcacaGLPaaacqGHsisl daqadaqaaiaahkgacqGHflY1caWHJbaacaGLOaGaayzkaaWaaeWaae aacaWHHbGaeyyXICTaaCizaaGaayjkaiaawMcaaaaa@7003@

 

2. The stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain relation for linear elasticity may be expressed as

σ ij = E 1+ν ( ε ij + ν 12ν ε kk δ ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUcaRiab e27aUbaadaqadaqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqaba GccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaIYaGa eqyVd4gaaiabew7aLnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0o azdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaaaaa@4F4E@

where σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3691@  and ε ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3675@  are the components of the stress and strain tensor, and E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbaaaa@338F@  and ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347D@  denote Young’s modulus and Poisson’s ratio.  Find an expression for strain in terms of stress.

 

Set i=j to see that

σ ii = E 1+ν ( ε ii + ν 12ν ε kk δ ii ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadM gaaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUcaRiab e27aUbaadaqadaqaaiabew7aLnaaBaaaleaacaWGPbGaamyAaaqaba GccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaIYaGa eqyVd4gaaiabew7aLnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0o azdaWgaaWcbaGaamyAaiaadMgaaeqaaaGccaGLOaGaayzkaaaaaa@4F4B@

Recall that δ ii =3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazdaWgaaWcbaGaamyAaiaadM gaaeqaaOGaeyypa0JaaG4maaaa@383F@ , and notice that we can replace the remaining ii by kk

σ kk = E 1+ν ( ε kk + ν 12ν 3 ε kk )= E 12ν ε kk ε kk = 12ν E σ kk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGRb Gaam4AaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4k aSIaeqyVd4gaamaabmaabaGaeqyTdu2aaSbaaSqaaiaadUgacaWGRb aabeaakiabgUcaRmaalaaabaGaeqyVd4gabaGaaGymaiabgkHiTiaa ikdacqaH9oGBaaGaaG4maiabew7aLnaaBaaaleaacaWGRbGaam4Aaa qabaaakiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadweaaeaacaaI XaGaeyOeI0IaaGOmaiabe27aUbaacqaH1oqzdaWgaaWcbaGaam4Aai aadUgaaeqaaaGcbaGaeyi1HSTaaGPaVlaaykW7caaMc8UaeqyTdu2a aSbaaSqaaiaadUgacaWGRbaabeaakiabg2da9maalaaabaGaaGymai abgkHiTiaaikdacqaH9oGBaeaacaWGfbaaaiabeo8aZnaaBaaaleaa caWGRbGaam4Aaaqabaaaaaa@6A9E@

Now, substitute for ε kk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaam4AaiaadU gaaeqaaaaa@3678@  in the given stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain relation

σ ij = E 1+ν ( ε ij + ν E σ kk δ ij ) ε ij = 1+ν E ( σ ij ν 1+ν σ kk δ ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4k aSIaeqyVd4gaamaabmaabaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQb aabeaakiabgUcaRmaalaaabaGaeqyVd4gabaGaamyraaaacqaHdpWC daWgaaWcbaGaam4AaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadM gacaWGQbaabeaaaOGaayjkaiaawMcaaaqaaiabgsDiBlabew7aLnaa BaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiaaigdacq GHRaWkcqaH9oGBaeaacaWGfbaaamaabmaabaGaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaakiabgkHiTmaalaaabaGaeqyVd4gabaGaaG ymaiabgUcaRiabe27aUbaacqaHdpWCdaWgaaWcbaGaam4AaiaadUga aeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkai aawMcaaaaaaa@6A64@

 

3. Solve the equation

 

μ{ δ kj a i a i + 1 12ν a k a j } U k = P j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBdaGadaqaaiabes7aKnaaBa aaleaacaWGRbGaamOAaaqabaGccaWGHbWaaSbaaSqaaiaadMgaaeqa aOGaamyyamaaBaaaleaacaWGPbaabeaakiabgUcaRmaalaaabaGaaG ymaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaaiaadggadaWgaaWc baGaam4AaaqabaGccaWGHbWaaSbaaSqaaiaadQgaaeqaaaGccaGL7b GaayzFaaGaamyvamaaBaaaleaacaWGRbaabeaakiabg2da9iaadcfa daWgaaWcbaGaamOAaaqabaaaaa@4D50@

for U k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadUgaaeqaaa aa@34BB@  in terms of P i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaa aa@34B4@  and a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaa aa@34C5@

 

Multiply both sides by a j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa WcbaGaamOAaaqabaaaaa@383A@  to see that

μ{ a j δ kj a i a i + 1 12ν a k a j a j } U k = P j a j μ{ a k a i a i + 1 12ν a k a j a j } U k = P j a j μ U k a k 2( 1ν ) 12ν a i a i = P j a j U k a k = (12ν) P j a j 2μ( 1ν ) a i a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeY7aTnaacmaabaGaamyyam aaBaaaleaacaWGQbaabeaakiabes7aKnaaBaaaleaacaWGRbGaamOA aaqabaGccaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaamyyamaaBaaale aacaWGPbaabeaakiabgUcaRmaalaaabaGaaGymaaqaaiaaigdacqGH sislcaaIYaGaeqyVd4gaaiaadggadaWgaaWcbaGaam4AaaqabaGcca WGHbWaaSbaaSqaaiaadQgaaeqaaOGaamyyamaaBaaaleaacaWGQbaa beaaaOGaay5Eaiaaw2haaiaadwfadaWgaaWcbaGaam4AaaqabaGccq GH9aqpcaWGqbWaaSbaaSqaaiaadQgaaeqaaOGaamyyamaaBaaaleaa caWGQbaabeaaaOqaaiabgsDiBlaaykW7caaMc8UaeqiVd02aaiWaae aacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWG PbaabeaakiaadggadaWgaaWcbaGaamyAaaqabaGccqGHRaWkdaWcaa qaaiaaigdaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacaWGHbWa aSbaaSqaaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWGQbaabeaaki aadggadaWgaaWcbaGaamOAaaqabaaakiaawUhacaGL9baacaWGvbWa aSbaaSqaaiaadUgaaeqaaOGaeyypa0JaamiuamaaBaaaleaacaWGQb aabeaakiaadggadaWgaaWcbaGaamOAaaqabaaakeaacqGHuhY2cqaH 8oqBcaWGvbWaaSbaaSqaaiaadUgaaeqaaOGaamyyamaaBaaaleaaca WGRbaabeaakmaalaaabaGaaGOmamaabmaabaGaaGymaiabgkHiTiab e27aUbGaayjkaiaawMcaaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4 gaaiaadggadaWgaaWcbaGaamyAaaqabaGccaWGHbWaaSbaaSqaaiaa dMgaaeqaaOGaeyypa0JaamiuamaaBaaaleaacaWGQbaabeaakiaadg gadaWgaaWcbaGaamOAaaqabaGccaaMc8UaaGPaVlaaykW7cqGHuhY2 caWGvbWaaSbaaSqaaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWGRb aabeaakiabg2da9maalaaabaGaaiikaiaaigdacqGHsislcaaIYaGa eqyVd4MaaiykaiaadcfadaWgaaWcbaGaamOAaaqabaGccaWGHbWaaS baaSqaaiaadQgaaeqaaaGcbaGaaGOmaiabeY7aTnaabmaabaGaaGym aiabgkHiTiabe27aUbGaayjkaiaawMcaaiaadggadaWgaaWcbaGaam yAaaqabaGccaWGHbWaaSbaaSqaaiaadMgaaeqaaaaaaaaa@B1FE@

Substitute back into the equation given for U k a k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadUgaaeqaaO GaamyyamaaBaaaleaacaWGRbaabeaaaaa@36C7@  to see that

μ U j a i a i + P k a k 2(1ν) a i a i a j = P j U j = 1 μ a i a i ( P j P k a k 2(1ν) a n a n a j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBcaWGvbWaaSbaaSqaaiaadQ gaaeqaaOGaamyyamaaBaaaleaacaWGPbaabeaakiaadggadaWgaaWc baGaamyAaaqabaGccqGHRaWkdaWcaaqaaiaadcfadaWgaaWcbaGaam 4AaaqabaGccaWGHbWaaSbaaSqaaiaadUgaaeqaaaGcbaGaaGOmaiaa cIcacaaIXaGaeyOeI0IaeqyVd4MaaiykaiaadggadaWgaaWcbaGaam yAaaqabaGccaWGHbWaaSbaaSqaaiaadMgaaeqaaaaakiaadggadaWg aaWcbaGaamOAaaqabaGccqGH9aqpcaWGqbWaaSbaaSqaaiaadQgaae qaaOGaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaadwfadaWgaaWc baGaamOAaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacqaH8oqBca WGHbWaaSbaaSqaaiaadMgaaeqaaOGaamyyamaaBaaaleaacaWGPbaa beaaaaGcdaqadaqaaiaadcfadaWgaaWcbaGaamOAaaqabaGccqGHsi sldaWcaaqaaiaadcfadaWgaaWcbaGaam4AaaqabaGccaWGHbWaaSba aSqaaiaadUgaaeqaaaGcbaGaaGOmaiaacIcacaaIXaGaeyOeI0Iaeq yVd4MaaiykaiaadggadaWgaaWcbaGaamOBaaqabaGccaWGHbWaaSba aSqaaiaad6gaaeqaaaaakiaadggadaWgaaWcbaGaamOAaaqabaaaki aawIcacaGLPaaaaaa@7463@

 

4. Let r= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa aeqaaaaa@390F@ .  Calculate r x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadkhaaeaacq GHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaaaaa@38B0@

 

We can just apply the usual chain and product rules of differentiation

r x i = 1 2 1 x k x k ( x k x k x i + x k x i x k )= 1 x k x k x k δ ik = x i x k x k = x i r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadkhaaeaacq GHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaaaWaaSaaaeaacaaIXaaabaWaaOaaaeaaca WG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaa beaaaeqaaaaakmaabmaabaGaamiEamaaBaaaleaacaWGRbaabeaakm aalaaabaGaeyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaamyAaaqabaaaaOGaey4kaSYaaSaaae aacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaWGPbaabeaaaaGccaWG4bWaaSbaaSqaaiaadU gaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaWa aOaaaeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaale aacaWGRbaabeaaaeqaaaaakiaadIhadaWgaaWcbaGaam4AaaqabaGc cqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaOGaeyypa0ZaaSaaae aacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGcbaWaaOaaaeaacaWG4bWa aSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaaae qaaaaakiabg2da9maalaaabaGaamiEamaaBaaaleaacaWGPbaabeaa aOqaaiaadkhaaaaaaa@6D17@

 

5. Let λ= A ij A ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBcqGH9aqpcaWGbbWaaSbaaS qaaiaadMgacaWGQbaabeaakiaadgeadaWgaaWcbaGaamyAaiaadQga aeqaaaaa@3B28@ .  Calculate λ/ A kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaH7oaBcaGGVaGaeyOaIy RaamyqamaaBaaaleaacaWGRbGaamiBaaqabaaaaa@3ACC@

 

Using the product rule

λ A kl = A ij δ ik δ jl + δ ik δ jl A ij =2 A kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kabeU7aSbqaai abgkGi2kaadgeadaWgaaWcbaGaam4AaiaadYgaaeqaaaaakiabg2da 9iaadgeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeqiTdq2aaSbaaS qaaiaadMgacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGQbGaamiB aaqabaGccqGHRaWkcqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaO GaeqiTdq2aaSbaaSqaaiaadQgacaWGSbaabeaakiaadgeadaWgaaWc baGaamyAaiaadQgaaeqaaOGaeyypa0JaaGOmaiaadgeadaWgaaWcba Gaam4AaiaadYgaaeqaaaaa@554A@