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 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  be a Cartesian basis. Denote the components of x in this basis by ( x 1 , x 2 , x 3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaam iEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4maaqabaaakiaawI [email protected]@ , 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 ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaaeaqabe aacaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaacaaIXaGaaGOmaaqaba GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaSba aSqaaiaaigdacaaIZaaabeaaaOqaaiaadofadaWgaaWcbaGaaGOmai aaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWa aSbaaSqaaiaaikdacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGc baGaam4uamaaBaaaleaacaaIZaGaaGymaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaadofadaWgaaWcbaGaaG4maiaaikdaaeqa aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaBa [email protected]@

Using index notation, we would express x and S as

x x i S S ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahIhacqGHHj IUcaWG4bWaaSbaaSqaaiaadMgacaaMc8oabeaakiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaC4uaiabggMi6kaadofadaWgaaWc [email protected]@

 

2.2. Conventions and special symbols for index notation

 

 Range Convention: Lower case Latin subscripts (i, j, k…) have the range ( 1,2,3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG [email protected]@ .  The symbol x i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa [email protected]@  denotes three components of a vector x 1 , x 2 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccaGGSaGaaGPaVlaaykW7caWG4bWaaSbaaSqa [email protected]@  and x 3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa [email protected]@ .  The symbol S ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaa [email protected]@  denotes nine components of a second order tensor, S 11 , S 12 , S 13 , S 21 S 33 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaa WcbaGaaGymaiaaigdaaeqaaOGaaiilaiaaykW7caaMc8UaaGPaVlaa ykW7caWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaacYcacaaMc8 UaaGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaacaaIXaGaaG4maaqa baGccaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaS baaSqaaiaaikdacaaIXaaabeaakiaaykW7caaMc8UaeSOjGSKaaGPa [email protected]@

 

 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 ] [email protected]@[email protected]@+= 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 [email protected]@

 

In the last two equations, [ A ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam [email protected]@ , [ B ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam [email protected]@  and [ C ] [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaGaam [email protected]@  denote the ( 3×3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG [email protected]@  component matrices of A, B and C.

 

 The Kronecker Delta:  The symbol δ ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa [email protected]@  is known as the Kronecker delta, and has the properties

δ ij ={ 1i=j 0ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaGabaabaeqabaGaaGym aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaamyAaiabg2da9iaadQgaaeaacaaIWaGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca [email protected]@

thus

δ 11 = δ 22 = δ 33 =1 δ 12 = δ 21 = δ 13 = δ 31 = δ 32 = δ 32 =0 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaaIXaGaaGymaaqabaGccqGH9aqpcqaH0oazdaWgaaWcbaGa aGOmaiaaikdaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaaiodaca aIZaaabeaakiabg2da9iaaigdacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq aH0oazdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaeqiTdq2a aSbaaSqaaiaaikdacaaIXaaabeaakiabg2da9iabes7aKnaaBaaale aacaaIXaGaaG4maaqabaGccqGH9aqpcqaH0oazdaWgaaWcbaGaaG4m aiaaigdaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaaiodacaaIYa aabeaakiabg2da9iabes7aKnaaBaaaleaacaaIZaGaaGOmaaqabaGc [email protected]@

You can also think of δ ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa [email protected]@  as the components of the identity tensor, or a ( 3×3 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaG [email protected]@  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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqiTdq 2aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iabes7aKnaaBaaa leaacaWGQbGaamyAaaqabaaakeaacqaH0oazdaWgaaWcbaGaam4Aai aadUgaaeqaaOGaeyypa0JaaG4maaqaaiaadggadaWgaaWcbaGaamyA aaqabaGccqGH9aqpcqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaO GaamyyamaaBaaaleaacaWGRbaabeaaaOqaaiaadgeadaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgaca [email protected]@

 

 The Permutation Symbol: The symbol ijk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgIGiopaaBa [email protected]@  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 [email protected]@[email protected]@+= 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 [email protected]@

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 [email protected]@[email protected]@+= 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 [email protected]@

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 ) [email protected]@[email protected]@+= 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 [email protected]@

 

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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaamyAaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa kiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaamyqamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGcbWaaSbaaSqa aiaadUgacaWGQbaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGbbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaO [email protected]@

are valid, but

A kk x k , A ik B kk , A ij B ik C ik [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaa WcbaGaam4AaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa kiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaamyqamaaBaaaleaacaWGPbGaam4AaaqabaGccaWGcbWaaSbaaSqa aiaadUgacaWGRbaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGbbWaaSbaaSqaaiaa dMgacaWGQbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaO [email protected]@

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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEam aaBaaaleaacaWGPbaabeaakiabg2da9iaadwhadaWgaaWcbaGaamyA aaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eyyyIORaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aahIhacqGH9aqpcaWH1bGaey4kaSIaaC4yaaqaaiaadggadaWgaaWc baGaamyAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaadUgacaWGPb aabeaakiaadkeadaWgaaWcbaGaam4AaiaadQgaaeqaaOGaamiEamaa BaaaleaacaWGQbaabeaakiabgUcaRiaadoeadaWgaaWcbaGaamyAai aadUgaaeqaaOGaamyDamaaBaaaleaacaWGRbaabeaakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaeyyyIORaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaCyyaiabg2da9iaahgeadaahaaWcbeqaaiaadsfaaaGccaWHcb [email protected]@

are valid, but

x i = A ij x j = A ik u k x i = A ik u k + c j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEam aaBaaaleaacaWGPbaabeaakiabg2da9iaadgeadaWgaaWcbaGaamyA aiaadQgaaeqaaaGcbaGaamiEamaaBaaaleaacaWGQbaabeaakiabg2 da9iaadgeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamyDamaaBaaa leaacaWGRbaabeaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWGbbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadwhadaWg aaWcbaGaam4AaaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaadQgaae [email protected]@

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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaamyAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaaiaadMgacaWG RbaabeaakiaadIhadaWgaaWcbaGaam4AaaqabaGccqGHuhY2caWG4b WaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamyqamaaBaaaleaacaWG QbGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaeyi1HS TaamiEamaaBaaaleaacaWGQbaabeaakiabg2da9iaadgeadaWgaaWc baGaamOAaiaadMgaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaaaa [email protected]@

 

2.4. Vector operations expressed using index notation

 

 Addition.   c=a+b c i = a i + b i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogacqGH9a qpcaWHHbGaey4kaSIaaCOyaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaadogadaWgaaWcbaGaamyAaaqabaGc cqGH9aqpcaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamOyam [email protected]@

 

 Dot Product  λ=abλ= a i b i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSjabg2 da9iaahggacqGHflY1caWHIbGaaGPaVlaaykW7caaMc8UaaGPaVlab ggMi6kaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4UdWMaeyypa0 JaamyyamaaBaaaleaacaWGPbaabeaakiaadkgadaWgaaWcbaGaamyA [email protected]@

 

 Vector Product c=a×b c i = ijk a j b k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahogacqGH9a qpcaWHHbGaey41aqRaaCOyaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyyyIORaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG JbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGPaVlabgIGiopaaBa aaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWG [email protected]@

 

 Dyadic Product   S=ab S ij = a i b j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahofacqGH9a qpcaWHHbGaey4LIqSaaCOyaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtbWaaSbaaSqaaiaadMga caWGQbaabeaakiabg2da9iaadggadaWgaaWcbaGaamyAaaqabaGcca [email protected]@

 

 Change of Basis.  Let a be a vector. Let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  be a Cartesian basis, and denote the components of a in this basis by a i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa [email protected]@ .  Let { m 1 , m 2 , m 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yBamaaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  be a second basis, and denote the components of a in this basis by α i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqydaWgaaWcbaGaamyAaaqaba [email protected]@ .  Then, define

Q ij = m i e j =cosθ( m i , e j ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaCyBamaaBaaaleaacaWG PbaabeaakiabgwSixlaahwgadaWgaaWcbaGaamOAaaqabaGccqGH9a qpciGGJbGaai4BaiaacohacqaH4oqCcaGGOaGaaCyBamaaBaaaleaa caWGPbaabeaakiaacYcacaWHLbWaaSbaaSqaaiaadQgaaeqaaOGaai [email protected]@

where θ( m i , e j ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaacI cacaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaahwgadaWgaaWc [email protected]@  denotes the angle between the unit vectors m i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa [email protected]@   and e j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa [email protected]@ .  Then

α i = Q ij a j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHnaaBa aaleaacaWGPbaabeaakiabg2da9iaadgfadaWgaaWcbaGaamyAaiaa [email protected]@

 

2.5. Tensor operations expressed using index notation.

 

 Addition.   C=A+B C ij = A ij + B ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbGaeyypa0JaaCyqaiabgUcaRi aahkeacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7cqGHHjIUcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWGdbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaa dgeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaamOqamaaBa [email protected]@

 

 Transpose  A= B T A ij = B ji [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHbbGaeyypa0JaaCOqamaaCaaale qabaGaamivaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyyyIORaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadgeadaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyypa0JaamOqamaaBaaaleaacaWGQbGaam [email protected]@

 

 Scalar Products λ=A:Bλ= A ij B ij λ=ABλ= A ji B ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeU7aSjabg2da9iaahgeaca GG6aGaaCOqaiaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaeq4UdWMaeyypa0JaamyqamaaBaaaleaaca WGPbGaamOAaaqabaGccaWGcbWaaSbaaSqaaiaadMgacaWGQbaabeaa aOqaaiabeU7aSjabg2da9iaahgeacqGHflY1cqGHflY1caaMc8UaaC OqaiaaykW7caaMc8UaaGPaVlaaykW7cqGHHjIUcqaH7oaBcqGH9aqp caWGbbWaaSbaaSqaaiaadQgacaWGPbaabeaakiaadkeadaWgaaWcba [email protected]@

 

 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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahogacqGH9aqpcaWHbbGaaC OyaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlabggMi6kaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGJbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Jaamyq amaaBaaaleaacaWGPbGaamOAaaqabaGccaWGIbWaaSbaaSqaaiaadQ gaaeqaaaGcbaGaaC4yaiabg2da9iaahgeadaahaaWcbeqaaiaadsfa aaGccaWHIbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabggMi6kaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa dogadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGbbWaaSbaaSqaai [email protected]@

 

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

* 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 [email protected]@[email protected]@+= 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 [email protected]@

* Inverse S ji 1 = 1 2det(S) ipq jkl S pk S ql [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGtbWaa0baaS qaaiaadQgacaWGPbaabaGaeyOeI0IaaGymaaaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaciGGKbGaaiyzaiaacshacaGGOaGaaC4uai aacMcaaaGaeyicI48aaSbaaSqaaiaadMgacaWGWbGaamyCaaqabaGc cqGHiiIZdaWgaaWcbaGaamOAaiaadUgacaWGSbaabeaakiaadofada WgaaWcbaGaamiCaiaadUgaaeqaaOGaam4uamaaBaaaleaacaWGXbGa [email protected]@

 Change of Basis.  Let A be a second order tensor. Let { e 1 , e 2 , e 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaaC yzamaaBaaaleaacaaIXaaabeaakiaacYcacaWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaiilaiaahwgadaWgaaWcbaGaaG4maaqabaaakiaawU [email protected]@  be a Cartesian basis, and denote the components of A in this basis by A ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgacaWGQb [email protected]@ .  Let { m 1 , m 2 , m 3 } [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca [email protected]@  be a second basis, and denote the components of A in this basis by Λ ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfU5amnaaBa [email protected]@ .  Then, define

Q ij = m i e j =cosθ( m i , e j ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgfadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaCyBamaaBaaaleaacaWG PbaabeaakiabgwSixlaahwgadaWgaaWcbaGaamOAaaqabaGccqGH9a qpciGGJbGaai4BaiaacohacqaH4oqCcaGGOaGaaCyBamaaBaaaleaa caWGPbaabeaakiaacYcacaWHLbWaaSbaaSqaaiaadQgaaeqaaOGaai [email protected]@

where θ( m i , e j ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeI7aXjaacI cacaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiaahwgadaWgaaWc [email protected]@  denotes the angle between the unit vectors m i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaah2gadaWgaa [email protected]@   and e j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa [email protected]@ .  Then

Λ ij = Q ik A km Q jm [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfU5amnaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGrbWaaSbaaSqaaiaa dMgacaWGRbaabeaakiaadgeadaWgaaWcbaGaam4Aaiaad2gaaeqaaO [email protected]@

 

 

2.6. Calculus using index notation

 

The derivative x i / x j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaa [email protected]@  can be deduced by noting that x i / x j =1i=j [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc cqGH9aqpcaaIXaGaaGPaVlaaykW7caaMc8UaaGPaVlaadMgacqGH9a [email protected]@  and  x i / x j =0ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc cqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaadMgacqGHGj [email protected]@ .  Therefore

                                                                  x i x j = δ ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadIhadaWgaa WcbaGaamyAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaaaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqaba [email protected]@

The same argument can be used for higher order tensors

                                                               A ij A kl = δ ik δ jl [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadgeadaWgaa WcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRaamyqamaaBaaaleaa caWGRbGaamiBaaqabaaaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadM gacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGQbGaamiBaaqabaaa [email protected]@

 

 

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 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahggacqGHxdaTcaWHIb aacaGLOaGaayzkaaGaeyyXIC9aaeWaaeaacaWHJbGaey41aqRaaCiz aaGaayjkaiaawMcaaiabg2da9maabmaabaGaaCyyaiabgwSixlaaho gaaiaawIcacaGLPaaadaqadaqaaiaahkgacqGHflY1caWHKbaacaGL OaGaayzkaaGaeyOeI0YaaeWaaeaacaWHIbGaeyyXICTaaC4yaaGaay jkaiaawMcaamaabmaabaGaaCyyaiabgwSixlaahsgaaiaawIcacaGL [email protected]@

 

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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahggacqGHxdaTcaWHIb aacaGLOaGaayzkaaGaeyyXIC9aaeWaaeaacaWHJbGaey41aqRaaCiz aaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7cqGHHjIUca aMc8UaaGPaVlaaykW7caaMc8UaeyicI48aaSbaaSqaaiaadMgacaWG QbGaam4AaaqabaGccaWGHbWaaSbaaSqaaiaadQgaaeqaaOGaamOyam aaBaaaleaacaWGRbaabeaakiabgIGiopaaBaaaleaacaWGPbGaamyB aiaad6gaaeqaaOGaam4yamaaBaaaleaacaWGTbaabeaakiaadsgada [email protected]@

Recall the identity

ijk imn = δ jm δ kn δ jn δ mk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQ gacaWGRbaabeaakiabgIGiopaaBaaaleaacaWGPbGaamyBaiaad6ga aeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadQgacaWGTbaabeaaki abes7aKnaaBaaaleaacaWGRbGaamOBaaqabaGccqGHsislcqaH0oaz daWgaaWcbaGaamOAaiaad6gaaeqaaOGaeqiTdq2aaSbaaSqaaiaad2 [email protected]@

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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlabgIGiopaaBaaale aacaWGPbGaamOAaiaadUgaaeqaaOGaamyyamaaBaaaleaacaWGQbaa beaakiaadkgadaWgaaWcbaGaam4AaaqabaGccqGHiiIZdaWgaaWcba GaamyAaiaad2gacaWGUbaabeaakiaadogadaWgaaWcbaGaamyBaaqa baGccaWGKbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0ZaaeWaaeaacq aH0oazdaWgaaWcbaGaamOAaiaad2gaaeqaaOGaeqiTdq2aaSbaaSqa aiaadUgacaWGUbaabeaakiabgkHiTiabes7aKnaaBaaaleaacaWGQb GaamOBaaqabaGccqaH0oazdaWgaaWcbaGaamyBaiaadUgaaeqaaaGc caGLOaGaayzkaaGaamyyamaaBaaaleaacaWGQbaabeaakiaadkgada WgaaWcbaGaam4AaaqabaGccaWGJbWaaSbaaSqaaiaad2gaaeqaaOGa [email protected]@

Multiply out, and note that

δ jm a j = a m δ kn b k = b n [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazdaWgaaWcbaGaamOAaiaad2 gaaeqaaOGaamyyamaaBaaaleaacaWGQbaabeaakiabg2da9iaadgga daWgaaWcbaGaamyBaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH 0oazdaWgaaWcbaGaam4Aaiaad6gaaeqaaOGaamOyamaaBaaaleaaca [email protected]@

(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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlaaykW7daqadaqaai abes7aKnaaBaaaleaacaWGQbGaamyBaaqabaGccqaH0oazdaWgaaWc baGaam4Aaiaad6gaaeqaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadQ gacaWGUbaabeaakiabes7aKnaaBaaaleaacaWGTbGaam4Aaaqabaaa kiaawIcacaGLPaaacaWGHbWaaSbaaSqaaiaadQgaaeqaaOGaamOyam aaBaaaleaacaWGRbaabeaakiaadogadaWgaaWcbaGaamyBaaqabaGc caWGKbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamyyamaaBaaale aacaWGTbaabeaakiaadkgadaWgaaWcbaGaamOBaaqabaGccaWGJbWa aSbaaSqaaiaad2gaaeqaaOGaamizamaaBaaaleaacaWGUbaabeaaki abgkHiTiaadggadaWgaaWcbaGaamOBaaqabaGccaWGIbWaaSbaaSqa aiaad2gaaeqaaOGaam4yamaaBaaaleaacaWGTbaabeaakiaadsgada [email protected]@

Finally, note that

a m c m ac [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaad2gaaeqaaO Gaam4yamaaBaaaleaacaWGTbaabeaakiaaykW7caaMc8UaaGPaVlaa [email protected]@

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 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaad2gaaeqaaO GaamOyamaaBaaaleaacaWGUbaabeaakiaadogadaWgaaWcbaGaamyB aaqabaGccaWGKbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0Iaamyyam aaBaaaleaacaWGUbaabeaakiaadkgadaWgaaWcbaGaamyBaaqabaGc caWGJbWaaSbaaSqaaiaad2gaaeqaaOGaamizamaaBaaaleaacaWGUb aabeaakiabg2da9iaadggadaWgaaWcbaGaamyBaaqabaGccaWGJbWa aSbaaSqaaiaad2gaaeqaaOGaamOyamaaBaaaleaacaWGUbaabeaaki aadsgadaWgaaWcbaGaamOBaaqabaGccqGHsislcaWGIbWaaSbaaSqa aiaad2gaaeqaaOGaam4yamaaBaaaleaacaWGTbaabeaakiaadggada WgaaWcbaGaamOBaaqabaGccaWGKbWaaSbaaSqaaiaad6gaaeqaaOGa eyyyIO7aaeWaaeaacaWHHbGaeyyXICTaaC4yaaGaayjkaiaawMcaam aabmaabaGaaCOyaiabgwSixlaahsgaaiaawIcacaGLPaaacqGHsisl daqadaqaaiaahkgacqGHflY1caWHJbaacaGLOaGaayzkaaWaaeWaae [email protected]@

 

2. The stress [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@ strain relation for linear elasticity may be expressed as

σ ij = E 1+ν ( ε ij + ν 12ν ε kk δ ij ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUcaRiab e27aUbaadaqadaqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqaba GccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaIYaGa eqyVd4gaaiabew7aLnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0o [email protected]@

where σ ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ [email protected]@  and ε ij [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamyAaiaadQ [email protected]@  are the components of the stress and strain tensor, and E [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  and ν [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa [email protected]@  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 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadM gaaeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGymaiabgUcaRiab e27aUbaadaqadaqaaiabew7aLnaaBaaaleaacaWGPbGaamyAaaqaba GccqGHRaWkdaWcaaqaaiabe27aUbqaaiaaigdacqGHsislcaaIYaGa eqyVd4gaaiabew7aLnaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0o [email protected]@

Recall that δ ii =3 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazdaWgaaWcbaGaamyAaiaadM [email protected]@ , 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 [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGRb Gaam4AaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4k aSIaeqyVd4gaamaabmaabaGaeqyTdu2aaSbaaSqaaiaadUgacaWGRb aabeaakiabgUcaRmaalaaabaGaeqyVd4gabaGaaGymaiabgkHiTiaa ikdacqaH9oGBaaGaaG4maiabew7aLnaaBaaaleaacaWGRbGaam4Aaa qabaaakiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadweaaeaacaaI XaGaeyOeI0IaaGOmaiabe27aUbaacqaH1oqzdaWgaaWcbaGaam4Aai aadUgaaeqaaaGcbaGaeyi1HSTaaGPaVlaaykW7caaMc8UaeqyTdu2a aSbaaSqaaiaadUgacaWGRbaabeaakiabg2da9maalaaabaGaaGymai abgkHiTiaaikdacqaH9oGBaeaacaWGfbaaaiabeo8aZnaaBaaaleaa [email protected]@

Now, substitute for ε kk [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaam4AaiaadU [email protected]@  in the given stress [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa [email protected]@ 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 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4k aSIaeqyVd4gaamaabmaabaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQb aabeaakiabgUcaRmaalaaabaGaeqyVd4gabaGaamyraaaacqaHdpWC daWgaaWcbaGaam4AaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadM gacaWGQbaabeaaaOGaayjkaiaawMcaaaqaaiabgsDiBlabew7aLnaa BaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiaaigdacq GHRaWkcqaH9oGBaeaacaWGfbaaamaabmaabaGaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaakiabgkHiTmaalaaabaGaeqyVd4gabaGaaG ymaiabgUcaRiabe27aUbaacqaHdpWCdaWgaaWcbaGaam4AaiaadUga aeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkai [email protected]@

 

3. Solve the equation

 

μ{ δ kj a i a i + 1 12ν a k a j } U k = P j [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBdaGadaqaaiabes7aKnaaBa aaleaacaWGRbGaamOAaaqabaGccaWGHbWaaSbaaSqaaiaadMgaaeqa aOGaamyyamaaBaaaleaacaWGPbaabeaakiabgUcaRmaalaaabaGaaG ymaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaaiaadggadaWgaaWc baGaam4AaaqabaGccaWGHbWaaSbaaSqaaiaadQgaaeqaaaGccaGL7b GaayzFaaGaamyvamaaBaaaleaacaWGRbaabeaakiabg2da9iaadcfa [email protected]@

for U k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadUgaaeqaaa [email protected]@  in terms of P i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaa [email protected]@  and a i [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaadMgaaeqaaa [email protected]@

 

Multiply both sides by a j MathType[email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadggadaWgaa [email protected]@  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 [email protected]@[email protected]@+= 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 [email protected]@

Substitute back into the equation given for U k a k [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGvbWaaSbaaSqaaiaadUgaaeqaaO [email protected]@  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 ) [email protected]@[email protected]@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBcaWGvbWaaSbaaSqaaiaadQ gaaeqaaOGaamyyamaaBaaaleaacaWGPbaabeaakiaadggadaWgaaWc baGaamyAaaqabaGccqGHRaWkdaWcaaqaaiaadcfadaWgaaWcbaGaam 4AaaqabaGccaWGHbWaaSbaaSqaaiaadUgaaeqaaaGcbaGaaGOmaiaa cIcacaaIXaGaeyOeI0IaeqyVd4MaaiykaiaadggadaWgaaWcbaGaam yAaaqabaGccaWGHbWaaSbaaSqaaiaadMgaaeqaaaaakiaadggadaWg aaWcbaGaamOAaaqabaGccqGH9aqpcaWGqbWaaSbaaSqaaiaadQgaae qaaOGaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaadwfadaWgaaWc baGaamOAaaqabaGccqGH9aqpdaWcaaqaaiaaigdaaeaacqaH8oqBca WGHbWaaSbaaSqaaiaadMgaaeqaaOGaamyyamaaBaaaleaacaWGPbaa beaaaaGcdaqadaqaaiaadcfadaWgaaWcbaGaamOAaaqabaGccqGHsi sldaWcaaqaaiaadcfadaWgaaWcbaGaam4AaaqabaGccaWGHbWaaSba aSqaaiaadUgaaeqaaaGcbaGaaGOmaiaacIcacaaIXaGaeyOeI0Iaeq yVd4MaaiykaiaadggadaWgaaWcbaGaamOBaaqabaGccaWGHbWaaSba aSqaaiaad6gaaeqaaaaakiaadggadaWgaaWcbaGaamOAaaqabaaaki [email protected]@

 

4. Let r= x k x k [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0ZaaOaaaeaacaWG4b WaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaa [email protected]@ .  Calculate r x i [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadkhaaeaacq [email protected]@

 

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 [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadkhaaeaacq GHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaaaWaaSaaaeaacaaIXaaabaWaaOaaaeaaca WG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaa beaaaeqaaaaakmaabmaabaGaamiEamaaBaaaleaacaWGRbaabeaakm aalaaabaGaeyOaIyRaamiEamaaBaaaleaacaWGRbaabeaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaamyAaaqabaaaaOGaey4kaSYaaSaaae aacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaWGPbaabeaaaaGccaWG4bWaaSbaaSqaaiaadU gaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaWa aOaaaeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaale aacaWGRbaabeaaaeqaaaaakiaadIhadaWgaaWcbaGaam4AaaqabaGc cqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaOGaeyypa0ZaaSaaae aacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGcbaWaaOaaaeaacaWG4bWa aSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRbaabeaaae qaaaaakiabg2da9maalaaabaGaamiEamaaBaaaleaacaWGPbaabeaa [email protected]@

 

5. Let λ= A ij A ij [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBcqGH9aqpcaWGbbWaaSbaaS qaaiaadMgacaWGQbaabeaakiaadgeadaWgaaWcbaGaamyAaiaadQga [email protected]@ .  Calculate λ/ A kl [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaH7oaBcaGGVaGaeyOaIy [email protected]@

 

Using the product rule

λ A kl = A ij δ ik δ jl + δ ik δ jl A ij =2 A kl [email protected]@[email protected]@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kabeU7aSbqaai abgkGi2kaadgeadaWgaaWcbaGaam4AaiaadYgaaeqaaaaakiabg2da 9iaadgeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeqiTdq2aaSbaaS qaaiaadMgacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGQbGaamiB aaqabaGccqGHRaWkcqaH0oazdaWgaaWcbaGaamyAaiaadUgaaeqaaO GaeqiTdq2aaSbaaSqaaiaadQgacaWGSbaabeaakiaadgeadaWgaaWc baGaamyAaiaadQgaaeqaaOGaeyypa0JaaGOmaiaadgeadaWgaaWcba [email protected]@