Appendix B

 

A Brief Introduction to Tensors and Their Properties

 

 

 

B.1. BASIC PROPERTIES OF TENSORS

 

 

B.1.1 Examples of Tensors

 

The gradient of a vector field is a good example of a tensor.  Visualize a vector field: at every point in space, the field has a vector value u( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cYcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@3C49@ .  Let G=u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHhbGaeyypa0JaaCyDaiabgEPiel abgEGirlaaykW7aaa@3AB3@  represent the gradient of u.  By definition, G enables you to calculate the change in u when you move from a point x in space to a nearby point at x+dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaey4kaSIaamizaiaahIhaaa a@3692@ :

du=Gdx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyDaiabg2da9iaahEeacq GHflY1caWGKbGaaCiEaaaa@3AB6@

G is a second order tensor.  From this example, we see that when you multiply a vector by a tensor, the result is another vector. 

 

This is a general property of all second order tensors.  A tensor is a linear mapping of a vector onto another vector.  Two examples, together with the vectors they operate on, are:

 

 The stress tensor

t=nσ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH0bGaeyypa0JaaGPaVlaah6gacq GHflY1cqaHdpWCaaa@3B57@

where n is a unit vector normal to a surface, σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCaaa@3488@  is the stress tensor and t is the traction vector acting on the surface.

 

 The deformation gradient tensor

dw=Fdx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaC4Daiabg2da9iaahAeacq GHflY1caWGKbGaaCiEaaaa@3AB7@

where dx is an infinitesimal line element in an undeformed solid, and dw is the vector representing the deformed line element.

 

 

B.1.2 Matrix representation of a tensor

 

To evaluate and manipulate tensors, we express them as components in a basis, just as for vectors.  We can use the displacement gradient to illustrate how this is done.  Let u( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cYcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@3C49@  be a vector field, and let G=u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHhbGaeyypa0JaaCyDaiabgEGird aa@371F@  represent the gradient of u.  Recall the definition of G

du=Gdx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyDaiabg2da9iaahEeacq GHflY1caWGKbGaaCiEaaaa@3AB6@

Now, let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@  be a Cartesian basis, and express both du and dx as components.  Then, calculate the components of du in terms of dx using the usual rules of calculus

d u 1 = u 1 x 1 d x 1 + u 1 x 2 d x 2 + u 1 x 3 d x 3 d u 2 = u 2 x 1 d x 1 + u 2 x 2 d x 2 + u 2 x 3 d x 3 d u 3 = u 3 x 1 d x 1 + u 3 x 2 d x 2 + u 3 x 3 d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadsgacaWG1bWaaSbaaSqaai aaigdaaeqaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG1bWaaSbaaSqa aiaaigdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabe aaaaGccaWGKbGaamiEamaaBaaaleaacaaIXaaabeaakiabgUcaRmaa laaabaGaeyOaIyRaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiabgk Gi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaamizaiaadIhadaWg aaWcbaGaaGOmaaqabaGccqGHRaWkdaWcaaqaaiabgkGi2kaadwhada WgaaWcbaGaaGymaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaa iodaaeqaaaaakiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaGcba GaamizaiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqa aiabgkGi2kaadwhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHciITca WG4bWaaSbaaSqaaiaaigdaaeqaaaaakiaadsgacaWG4bWaaSbaaSqa aiaaigdaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaS qaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaa beaaaaGccaWGKbGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRm aalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIYaaabeaaaOqaaiab gkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaamizaiaadIhada WgaaWcbaGaaG4maaqabaaakeaacaWGKbGaamyDamaaBaaaleaacaaI Zaaabeaakiabg2da9maalaaabaGaeyOaIyRaamyDamaaBaaaleaaca aIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaa aOGaamizaiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkdaWcaa qaaiabgkGi2kaadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciIT caWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiaadsgacaWG4bWaaSbaaS qaaiaaikdaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSba aSqaaiaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIZa aabeaaaaGccaWGKbGaamiEamaaBaaaleaacaaIZaaabeaaaaaa@99A7@

We could represent this as a matrix product

[ d u 1 d u 2 d u 3 ]=[ u 1 x 1 u 1 x 2 u 1 x 3 u 2 x 1 u 2 x 2 u 2 x 3 u 3 x 1 u 3 x 2 u 3 x 3 ][ d x 1 d x 2 d x 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacaWGKb GaamyDamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG1bWaaSba aSqaaiaaikdaaeqaaaGcbaGaamizaiaadwhadaWgaaWcbaGaaG4maa qabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaeaafaqabeWadaaa baWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaigdaaeqaaaGcba GaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaaakeaadaWcaaqa aiabgkGi2kaadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHciITca WG4bWaaSbaaSqaaiaaikdaaeqaaaaaaOqaamaalaaabaGaeyOaIyRa amyDamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaa WcbaGaaG4maaqabaaaaaGcbaWaaSaaaeaacqGHciITcaWG1bWaaSba aSqaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXa aabeaaaaaakeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaaGOm aaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaaaO qaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIYaaabeaaaOqa aiabgkGi2kaadIhadaWgaaWcbaGaaG4maaqabaaaaaGcbaWaaSaaae aacqGHciITcaWG1bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIyRa amiEamaaBaaaleaacaaIXaaabeaaaaaakeaadaWcaaqaaiabgkGi2k aadwhadaWgaaWcbaGaaG4maaqabaaakeaacqGHciITcaWG4bWaaSba aSqaaiaaikdaaeqaaaaaaOqaamaalaaabaGaeyOaIyRaamyDamaaBa aaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaG4m aaqabaaaaaaaaOGaay5waiaaw2faamaadmaabaqbaeqabmqaaaqaai aadsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaadIha daWgaaWcbaGaaGOmaaqabaaakeaacaWGKbGaamiEamaaBaaaleaaca aIZaaabeaaaaaakiaawUfacaGLDbaaaaa@874C@

From this we see that G can be represented as a 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIZaGaey41aqRaaG4maaaa@3656@  matrix.  The elements of the matrix are known as the components of G in the basis { 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@ .  All second order tensors can be represented in this form.  For example, a general second order tensor S could be written as

S[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyyIO7aamWaaeaafaqabe WadaaabaGaam4uamaaBaaaleaacaaIXaGaaGymaaqabaaakeaacaWG tbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadofadaWgaaWcba GaaGymaiaaiodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIYaGaaGym aaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIYaaabeaaaOqaai aadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaam4uamaaBaaa leaacaaIZaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodaca aIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaiodaaeqaaaaa aOGaay5waiaaw2faaaaa@4E29@

You have probably already seen the matrix representation of stress and strain components in introductory courses.

 

Since S can be represented as a matrix, all operations that can be performed on a 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIZaGaey41aqRaaG4maaaa@3656@  matrix can also be performed on S.  Examples include sums and products, the transpose, inverse, and determinant.  One can also compute eigenvalues and eigenvectors for tensors, and thus define the log of a tensor, the square root of a tensor, etc.  These tensor operations are summarized below.

 

Note that the numbers S 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIXa aabeaaaaa@353F@ , S 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa aabeaaaaa@3540@ , … S 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaIZa aabeaaaaa@3543@  depend on the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@ , just as the components of a vector depend on the basis used to represent the vector.  However, just as the magnitude and direction of a vector are independent of the basis, so the properties of a tensor are independent of the basis.  That is to say, if S is a tensor and u is a vector, then the vector

v=Su MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaC4uaiabgwSixl aahwhaaaa@38EE@

has the same magnitude and direction, irrespective of the basis used to represent u, v, and S.

 

 

B.1.3 The difference between a matrix and a tensor

 

If a tensor is a matrix, why is a matrix not the same thing as a tensor?  Well, although you can multiply the three components of a vector u by any 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIZaGaey41aqRaaG4maaaa@3656@  matrix,

[ b 1 b 2 b 3 ]=[ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ][ u 1 u 2 u 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacaWGIb WaaSbaaSqaaiaaigdaaeqaaaGcbaGaamOyamaaBaaaleaacaaIYaaa beaaaOqaaiaadkgadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaay zxaaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaamyyamaaBaaaleaa caaIXaGaaGymaaqabaaakeaacaWGHbWaaSbaaSqaaiaaigdacaaIYa aabeaaaOqaaiaadggadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGa amyyamaaBaaaleaacaaIYaGaaGymaaqabaaakeaacaWGHbWaaSbaaS qaaiaaikdacaaIYaaabeaaaOqaaiaadggadaWgaaWcbaGaaGOmaiaa iodaaeqaaaGcbaGaamyyamaaBaaaleaacaaIZaGaaGymaaqabaaake aacaWGHbWaaSbaaSqaaiaaiodacaaIYaaabeaaaOqaaiaadggadaWg aaWcbaGaaG4maiaaiodaaeqaaaaaaOGaay5waiaaw2faamaadmaaba qbaeqabmqaaaqaaiaadwhadaWgaaWcbaGaaGymaaqabaaakeaacaWG 1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyDamaaBaaaleaacaaIZa aabeaaaaaakiaawUfacaGLDbaaaaa@5C59@

the resulting three numbers ( b 1 , b 2 , b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dkgadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3B09@  may or may not represent the components of a vector.  If they are the components of a vector, then the matrix represents the components of a tensor A, if not, then the matrix is just an ordinary old matrix.

 

 To check whether ( b 1 , b 2 , b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dkgadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3B09@  are the components of a vector, you need to check how ( b 1 , b 2 , b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dkgadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3B09@  change due to a change of basis.  That is to say, choose a new basis, calculate the new components of u in this basis, and calculate the new matrix in this basis (the new elements of the matrix will depend on how the matrix was defined.  The elements may or may not change MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  if they don’t, then the matrix cannot be the components of a tensor).  Then, evaluate the matrix product to find a new left hand side, say ( β 1 , β 2 , β 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiabek7aInaaBaaaleaaca aIXaaabeaakiaacYcacqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaGG SaGaeqOSdi2aaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaaaa@3D67@ .  If  ( β 1 , β 2 , β 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiabek7aInaaBaaaleaaca aIXaaabeaakiaacYcacqaHYoGydaWgaaWcbaGaaGOmaaqabaGccaGG SaGaeqOSdi2aaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaaaa@3D67@  are related to ( b 1 , b 2 , b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dkgadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3B09@  by the same transformation that was used to calculate the new components of u, then ( b 1 , b 2 , b 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOyamaaBaaaleaacaaIXa aabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa dkgadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3B09@  are the components of a vector, and, therefore, the matrix represents the components of a tensor.

 

 

B.1.4 Creating a tensor using a dyadic product of two vectors.

 

Let a and b be two vectors.  The dyadic product of a and b  is a second order tensor S denoted by

S=ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyypa0JaaCyyaiabgEPiel aahkgaaaa@3886@ .

with the property

Su=( ab )u=a(bu) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaaCyDaiabg2da9m aabmaabaGaaCyyaiabgEPielaahkgaaiaawIcacaGLPaaacqGHflY1 caWH1bGaeyypa0JaaCyyaiaacIcacaWHIbGaeyyXICTaaCyDaiaacM caaaa@481B@

for all vectors u.  (Clearly, this maps u onto a vector parallel to a with magnitude | a |( bu ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiaahggaaiaawEa7caGLiW oadaqadaqaaiaahkgacqGHflY1caWH1baacaGLOaGaayzkaaaaaa@3C8E@  )

 

The components of ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHHbGaey4LIqSaaCOyaaaa@36A4@  in a basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@  are

[ a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 b 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGHb WaaSbaaSqaaiaaigdaaeqaaOGaamOyamaaBaaaleaacaaIXaaabeaa aOqaaiaadggadaWgaaWcbaGaaGymaaqabaGccaWGIbWaaSbaaSqaai aaikdaaeqaaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaadkga daWgaaWcbaGaaG4maaqabaaakeaacaWGHbWaaSbaaSqaaiaaikdaae qaaOGaamOyamaaBaaaleaacaaIXaaabeaaaOqaaiaadggadaWgaaWc baGaaGOmaaqabaGccaWGIbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaam yyamaaBaaaleaacaaIYaaabeaakiaadkgadaWgaaWcbaGaaG4maaqa baaakeaacaWGHbWaaSbaaSqaaiaaiodaaeqaaOGaamOyamaaBaaale aacaaIXaaabeaaaOqaaiaadggadaWgaaWcbaGaaG4maaqabaGccaWG IbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyyamaaBaaaleaacaaIZa aabeaakiaadkgadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaayzx aaaaaa@5607@

 

Note that not all tensors can be constructed using a dyadic product of only two vectors (this is because ( ab )u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahggacqGHxkcXcaWHIb aacaGLOaGaayzkaaGaeyyXICTaaCyDaaaa@3B75@  always has to be parallel to a, and therefore the representation cannot map a vector onto an arbitrary vector).  However, if a, b, and c are three independent vectors (i.e. no two of them are parallel) then all tensors can be constructed as a sum of scalar multiples of the nine possible dyadic products of these vectors. 

 

 

 

B.2. OPERATIONS ON SECOND ORDER TENSORS

 

 Tensor components

 

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 let S be a second order tensor.  The components of S in { 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@  may be represented as a matrix

[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadofadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaam4uamaaBaaaleaacaaIXaGaaG4maa qabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIXaaabeaaaOqaaiaa dofadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaam4uamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaI XaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcba Gaam4uamaaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzx aaaaaa@4B84@

where

S 11 = e 1 ( S e 1 ), S 12 = e 1 ( S e 2 ), S 11 = e 1 ( S e 3 ), S 21 = e 2 ( S e 1 ), S 22 = e 2 ( S e 2 ), S 21 = e 2 ( S e 3 ), S 31 = e 3 ( S e 1 ), S 32 = e 3 ( S e 2 ), S 31 = e 3 ( S e 3 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadofadaWgaaWcbaGaaGymai aaigdaaeqaaOGaeyypa0JaaCyzamaaBaaaleaacaaIXaaabeaakiab gwSixpaabmaabaGaaC4uaiabgwSixlaahwgadaWgaaWcbaGaaGymaa qabaaakiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGtb WaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iaahwgadaWgaaWc baGaaGymaaqabaGccqGHflY1daqadaqaaiaahofacqGHflY1caWHLb WaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaaiilaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaadofadaWgaaWcbaGaaGymaiaaigdaaeqa aOGaeyypa0JaaCyzamaaBaaaleaacaaIXaaabeaakiabgwSixpaabm aabaGaaC4uaiabgwSixlaahwgadaWgaaWcbaGaaG4maaqabaaakiaa wIcacaGLPaaacaGGSaaabaGaam4uamaaBaaaleaacaaIYaGaaGymaa qabaGccqGH9aqpcaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaeyyXIC9a aeWaaeaacaWHtbGaeyyXICTaaCyzamaaBaaaleaacaaIXaaabeaaaO GaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadofadaWgaa WcbaGaaGOmaiaaikdaaeqaaOGaeyypa0JaaCyzamaaBaaaleaacaaI YaaabeaakiabgwSixpaabmaabaGaaC4uaiabgwSixlaahwgadaWgaa WcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGSaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGH 9aqpcaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaeyyXIC9aaeWaaeaaca WHtbGaeyyXICTaaCyzamaaBaaaleaacaaIZaaabeaaaOGaayjkaiaa wMcaaiaacYcaaeaacaWGtbWaaSbaaSqaaiaaiodacaaIXaaabeaaki abg2da9iaahwgadaWgaaWcbaGaaG4maaqabaGccqGHflY1daqadaqa aiaahofacqGHflY1caWHLbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOa GaayzkaaGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uamaaBaaaleaaca aIZaGaaGOmaaqabaGccqGH9aqpcaWHLbWaaSbaaSqaaiaaiodaaeqa aOGaeyyXIC9aaeWaaeaacaWHtbGaeyyXICTaaCyzamaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWGtbWaaSbaaSqaaiaaiodacaaIXaaabeaakiabg2da9iaa hwgadaWgaaWcbaGaaG4maaqabaGccqGHflY1daqadaqaaiaahofacq GHflY1caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaGa aiilaaaaaa@23DB@

 

The representation of a tensor in terms of its components can also be expressed in dyadic form as

S= j=1 3 i=1 3 S ij e i e j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahofacqGH9aqpdaaeWbqaamaaqahaba Gaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccaWHLbWaaSbaaSqa aiaadMgaaeqaaOGaey4LIqSaaCyzamaaBaaaleaacaWGQbaabeaaae aacaWGPbGaeyypa0JaaGymaaqaaiaaiodaa0GaeyyeIuoaaSqaaiaa dQgacqGH9aqpcaaIXaaabaGaaG4maaqdcqGHris5aaaa@46AB@

This representation is particularly convenient when using polar coordinates, as described in Appendix E.

 

 Addition

Let S and T be two tensors.  Then U=S+T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHvbGaeyypa0JaaC4uaiabgUcaRi aahsfaaaa@3744@  is also a tensor.

 

Denote the Cartesian components of U, S and T by matrices as defined above.  The components of U are then related to the components of S and T by

[ U 11 U 12 U 13 U 21 U 22 U 23 U 31 U 32 U 33 ]=[ S 11 + T 11 S 12 + T 12 S 13 + T 13 S 21 + T 21 S 22 + T 22 S 23 + T 23 S 31 + T 31 S 32 + T 32 S 33 + T 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGvb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadwfadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaamyvamaaBaaaleaacaaIXaGaaG4maa qabaaakeaacaWGvbWaaSbaaSqaaiaaikdacaaIXaaabeaaaOqaaiaa dwfadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaamyvamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaWGvbWaaSbaaSqaaiaaiodacaaI XaaabeaaaOqaaiaadwfadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcba GaamyvamaaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzx aaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaaca aIXaGaaGymaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaigdacaaI XaaabeaaaOqaaiaadofadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaey 4kaSIaamivamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWa aSbaaSqaaiaaigdacaaIZaaabeaakiabgUcaRiaadsfadaWgaaWcba GaaGymaiaaiodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIYaGaaGym aaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaikdacaaIXaaabeaaaO qaaiaadofadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaey4kaSIaamiv amaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaai aaikdacaaIZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGOmaiaa iodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIZaGaaGymaaqabaGccq GHRaWkcaWGubWaaSbaaSqaaiaaiodacaaIXaaabeaaaOqaaiaadofa daWgaaWcbaGaaG4maiaaikdaaeqaaOGaey4kaSIaamivamaaBaaale aacaaIZaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaI ZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaG4maiaaiodaaeqaaa aaaOGaay5waiaaw2faaaaa@840C@

 

 Product of a tensor and a vector

 

Let u be a vector and S a second order tensor.  Then

v=Su MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaC4uaiabgwSixl aahwhaaaa@38EE@

is a vector. 

 

Let ( u 1 , u 2 , u 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaadwhadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaamyDamaaBaaaleaacaaIYaaabeaakiaacYca caWG1bWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaaaa@3B72@  and ( v 1 , v 2 , v 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaadAhadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaamODamaaBaaaleaacaaIYaaabeaakiaacYca caWG2bWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaaaa@3B75@  denote the components of vectors u and v in a Cartesian basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@ , and denote the Cartesian components of S as described above.  Then

[ v 1 v 2 v 3 ]=[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ][ u 1 u 2 u 3 ]=[ S 11 u 1 + S 12 u 2 + S 13 u 3 S 21 u 1 + S 22 u 2 + S 23 u 3 S 31 u 1 + S 32 u 2 + S 33 u 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadeaaaeaacaWG2b WaaSbaaSqaaiaaigdaaeqaaaGcbaGaamODamaaBaaaleaacaaIYaaa beaaaOqaaiaadAhadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaay zxaaGaeyypa0ZaamWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaa caaIXaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYa aabeaaaOqaaiaadofadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGa am4uamaaBaaaleaacaaIYaGaaGymaaqabaaakeaacaWGtbWaaSbaaS qaaiaaikdacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaGOmaiaa iodaaeqaaaGcbaGaam4uamaaBaaaleaacaaIZaGaaGymaaqabaaake aacaWGtbWaaSbaaSqaaiaaiodacaaIYaaabeaaaOqaaiaadofadaWg aaWcbaGaaG4maiaaiodaaeqaaaaaaOGaay5waiaaw2faamaadmaaba qbaeqabmqaaaqaaiaadwhadaWgaaWcbaGaaGymaaqabaaakeaacaWG 1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyDamaaBaaaleaacaaIZa aabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqadeaa aeaacaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaadwhadaWgaa WcbaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaigdacaaI YaaabeaakiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGtb WaaSbaaSqaaiaaigdacaaIZaaabeaakiaadwhadaWgaaWcbaGaaG4m aaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIXaaabeaakiaadw hadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaa ikdacaaIYaaabeaakiaadwhadaWgaaWcbaGaaGOmaaqabaGccqGHRa WkcaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaakiaadwhadaWgaaWc baGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaIXaaabe aakiaadwhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSba aSqaaiaaiodacaaIYaaabeaakiaadwhadaWgaaWcbaGaaGOmaaqaba GccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadwha daWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaayzxaaaaaa@8C6C@

 

The product

v=uS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaCyDaiabgwSixl aahofaaaa@38EE@

is also a vector.  In component form

[ v 1 v 2 v 3 ]=[ u 1 u 2 u 3 ][ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ]=[ u 1 S 11 + u 2 S 21 + u 3 S 31 u 1 S 12 + u 2 S 22 + u 3 S 32 u 1 S 13 + u 2 S 23 + u 3 S 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqabmaaaeaacaWG2b WaaSbaaSqaaiaaigdaaeqaaaGcbaGaamODamaaBaaaleaacaaIYaaa beaaaOqaaiaadAhadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaay zxaaGaeyypa0ZaamWaaeaafaqabeqadaaabaGaamyDamaaBaaaleaa caaIXaaabeaaaOqaaiaadwhadaWgaaWcbaGaaGOmaaqabaaakeaaca WG1bWaaSbaaSqaaiaaiodaaeqaaaaaaOGaay5waiaaw2faamaadmaa baqbaeqabmWaaaqaaiaadofadaWgaaWcbaGaaGymaiaaigdaaeqaaa GcbaGaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWa aSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiaadofadaWgaaWcbaGaaG OmaiaaigdaaeqaaaGcbaGaam4uamaaBaaaleaacaaIYaGaaGOmaaqa baaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiaado fadaWgaaWcbaGaaG4maiaaigdaaeqaaaGcbaGaam4uamaaBaaaleaa caaIZaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaaiaaiodacaaIZa aabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaauaabeqadeaa aeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaam4uamaaBaaaleaaca aIXaGaaGymaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaaikdaaeqa aOGaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkcaWG1b WaaSbaaSqaaiaaiodaaeqaaOGaam4uamaaBaaaleaacaaIZaGaaGym aaqabaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaam4uamaaBa aaleaacaaIXaGaaGOmaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaa ikdaaeqaaOGaam4uamaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHRa WkcaWG1bWaaSbaaSqaaiaaiodaaeqaaOGaam4uamaaBaaaleaacaaI ZaGaaGOmaaqabaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaOGaam 4uamaaBaaaleaacaaIXaGaaG4maaqabaGccqGHRaWkcaWG1bWaaSba aSqaaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaIYaGaaG4maaqaba GccqGHRaWkcaWG1bWaaSbaaSqaaiaaiodaaeqaaOGaam4uamaaBaaa leaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzxaaaaaa@8C6C@

Observe that uSSu MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyyXICTaaC4uaiabgcMi5k aahofacqGHflY1caWH1baaaa@3CD4@  (unless S is symmetric).

 

 Product of two tensors

 

Let T and S be two second order tensors.  Then U=TS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHvbGaeyypa0JaaCivaiabgwSixl aahofaaaa@38AC@  is also a tensor.

 

Denote the components of U, S and T by 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIZaGaey41aqRaaG4maaaa@3656@  matrices.  Then,

[ U 11 U 12 U 13 U 21 U 22 U 23 U 31 U 32 U 33 ]=[ T 11 T 12 T 13 T 21 T 22 T 23 T 31 T 32 T 33 ][ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] =[ T 11 S 11 + T 12 S 21 + T 13 S 31 T 11 S 12 + T 12 S 22 + T 13 S 32 T 11 S 13 + T 12 S 23 + T 13 S 33 T 21 S 11 + T 22 S 21 + T 23 S 31 T 21 S 12 + T 22 S 22 + T 23 S 32 T 21 S 12 + T 22 S 22 + T 23 S 32 T 31 S 11 + T 32 S 21 + T 33 S 31 T 31 S 12 + T 32 S 22 + T 33 S 32 T 31 S 13 + T 32 S 23 + T 33 S 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaadmaabaqbaeqabmWaaaqaai aadwfadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaamyvamaaBaaa leaacaaIXaGaaGOmaaqabaaakeaacaWGvbWaaSbaaSqaaiaaigdaca aIZaaabeaaaOqaaiaadwfadaWgaaWcbaGaaGOmaiaaigdaaeqaaaGc baGaamyvamaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacaWGvbWaaS baaSqaaiaaikdacaaIZaaabeaaaOqaaiaadwfadaWgaaWcbaGaaG4m aiaaigdaaeqaaaGcbaGaamyvamaaBaaaleaacaaIZaGaaGOmaaqaba aakeaacaWGvbWaaSbaaSqaaiaaiodacaaIZaaabeaaaaaakiaawUfa caGLDbaacqGH9aqpdaWadaqaauaabeqadmaaaeaacaWGubWaaSbaaS qaaiaaigdacaaIXaaabeaaaOqaaiaadsfadaWgaaWcbaGaaGymaiaa ikdaaeqaaaGcbaGaamivamaaBaaaleaacaaIXaGaaG4maaqabaaake aacaWGubWaaSbaaSqaaiaaikdacaaIXaaabeaaaOqaaiaadsfadaWg aaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaamivamaaBaaaleaacaaIYa GaaG4maaqabaaakeaacaWGubWaaSbaaSqaaiaaiodacaaIXaaabeaa aOqaaiaadsfadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaamivam aaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzxaaWaamWa aeaafaqabeWadaaabaGaam4uamaaBaaaleaacaaIXaGaaGymaaqaba aakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadofa daWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaam4uamaaBaaaleaaca aIYaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIYaaa beaaaOqaaiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcbaGaam 4uamaaBaaaleaacaaIZaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqa aiaaiodacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaio daaeqaaaaaaOGaay5waiaaw2faaaqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8Uaeyypa0ZaamWaaeaafaqabeWadaaabaGaamivamaaBa aaleaacaaIXaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaigdacaaI XaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGymaiaaikdaaeqaaO Gaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkcaWGubWa aSbaaSqaaiaaigdacaaIZaaabeaakiaadofadaWgaaWcbaGaaG4mai aaigdaaeqaaaGcbaGaamivamaaBaaaleaacaaIXaGaaGymaaqabaGc caWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaadsfada WgaaWcbaGaaGymaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaIYaGa aGOmaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaigdacaaIZaaabe aakiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaamivamaa BaaaleaacaaIXaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaigdaca aIZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGymaiaaikdaaeqa aOGaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGHRaWkcaWGub WaaSbaaSqaaiaaigdacaaIZaaabeaakiaadofadaWgaaWcbaGaaG4m aiaaiodaaeqaaaGcbaGaamivamaaBaaaleaacaaIYaGaaGymaaqaba GccaWGtbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaadsfa daWgaaWcbaGaaGOmaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaIYa GaaGymaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaikdacaaIZaaa beaakiaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaaGcbaGaamivam aaBaaaleaacaaIYaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaigda caaIYaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGOmaiaaikdaae qaaOGaam4uamaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHRaWkcaWG ubWaaSbaaSqaaiaaikdacaaIZaaabeaakiaadofadaWgaaWcbaGaaG 4maiaaikdaaeqaaaGcbaGaamivamaaBaaaleaacaaIYaGaaGymaaqa baGccaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaads fadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaam4uamaaBaaaleaacaaI YaGaaGOmaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaikdacaaIZa aabeaakiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaamiv amaaBaaaleaacaaIZaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaig dacaaIXaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaG4maiaaikda aeqaaOGaam4uamaaBaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkca WGubWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadofadaWgaaWcbaGa aG4maiaaigdaaeqaaaGcbaGaamivamaaBaaaleaacaaIZaGaaGymaa qabaGccaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaa dsfadaWgaaWcbaGaaG4maiaaikdaaeqaaOGaam4uamaaBaaaleaaca aIYaGaaGOmaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaaiodacaaI ZaaabeaakiaadofadaWgaaWcbaGaaG4maiaaikdaaeqaaaGcbaGaam ivamaaBaaaleaacaaIZaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaa igdacaaIZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaG4maiaaik daaeqaaOGaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGHRaWk caWGubWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadofadaWgaaWcba GaaG4maiaaiodaaeqaaaaaaOGaay5waiaaw2faaaaaaa@2130@

Note that tensor products, like matrix products, are not commutative; i.e. TSST MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaeyyXICTaaC4uaiabgcMi5k aahofacqGHflY1caWHubaaaa@3C92@

 

 Transpose

 

Let S be a tensor.  The transpose of S is denoted by S T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahofadaahaaWcbeqaaiaadsfaaaaaaa@324A@  and is defined so that

u S T =Su MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyyXICTaaC4uamaaCaaale qabaGaamivaaaakiabg2da9iaahofacqGHflY1caWH1baaaa@3D23@

 

Denote the components of S by a 3x3 matrix.  The components of  S T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahofadaahaaWcbeqaaiaadsfaaaaaaa@324A@  are then

S T [ S 11 S 21 S 31 S 12 S 22 S 32 S 13 S 23 S 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbWaaWbaaSqabeaacaWGubaaaO GaeyyyIO7aamWaaeaafaqabeWadaaabaGaam4uamaaBaaaleaacaaI XaGaaGymaaqabaaakeaacaWGtbWaaSbaaSqaaiaaikdacaaIXaaabe aaaOqaaiaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaaGcbaGaam4u amaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWaaSbaaSqaai aaikdacaaIYaaabeaaaOqaaiaadofadaWgaaWcbaGaaG4maiaaikda aeqaaaGcbaGaam4uamaaBaaaleaacaaIXaGaaG4maaqabaaakeaaca WGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiaadofadaWgaaWc baGaaG4maiaaiodaaeqaaaaaaOGaay5waiaaw2faaaaa@4F39@

i.e. the rows and columns of the matrix are switched.


Note that, if A and B are two tensors, then

( AB ) T = B T A T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiaahgeacqGHflY1caWHcb aacaGLOaGaayzkaaWaaWbaaSqabeaacaWGubaaaOGaeyypa0JaaCOq amaaCaaaleqabaGaamivaaaakiabgwSixlaahgeadaahaaWcbeqaai aadsfaaaaaaa@4038@


 Trace

 

Let S be a tensor, and denote the components of S by a 3×3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIZaGaey41aqRaaG4maaaa@3656@  matrix.  The trace of S is denoted by tr(S) or trace(S), and can be computed by summing the diagonals of the matrix of components

trace( S )= S 11 + S 22 + S 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqG0bGaaeOCaiaabggacaqGJbGaae yzamaabmaabaGaaC4uaaGaayjkaiaawMcaaiabg2da9iaadofadaWg aaWcbaGaaGymaiaaigdaaeqaaOGaey4kaSIaam4uamaaBaaaleaaca aIYaGaaGOmaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodacaaI Zaaabeaaaaa@441A@

More formally, let { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@  be any Cartesian basis.  Then

trace( S )= e 1 S e 1 + e 2 S e 2 + e 3 S e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqG0bGaaeOCaiaabggacaqGJbGaae yzamaabmaabaGaaC4uaaGaayjkaiaawMcaaiabg2da9iaahwgadaWg aaWcbaGaaGymaaqabaGccqGHflY1caWHtbGaeyyXICTaaCyzamaaBa aaleaacaaIXaaabeaakiabgUcaRiaahwgadaWgaaWcbaGaaGOmaaqa baGccqGHflY1caWHtbGaeyyXICTaaCyzamaaBaaaleaacaaIYaaabe aakiabgUcaRiaahwgadaWgaaWcbaGaaG4maaqabaGccqGHflY1caWH tbGaeyyXICTaaCyzamaaBaaaleaacaaIZaaabeaaaaa@5818@

The trace of a tensor is an example of an invariant of the tensor MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you get the same value for trace(S) whatever basis you use to define the matrix of components of S.

 

 Contraction.

 

Inner Product: Let S and T be two second order tensors.  The inner product of S and T is a scalar, denoted by S:T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaaiOoaiaahsfaaaa@353C@ .  Represent S and T by their components in a basis.  Then

S:T= S 11 T 11 + S 12 T 12 + S 13 T 13 + S 21 T 21 + S 22 T 22 + S 23 T 23 + S 31 T 31 + S 32 T 32 + S 33 T 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahofacaGG6aGaaCivaiabg2 da9iaadofadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaamivamaaBaaa leaacaaIXaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaig dacaaIYaaabeaakiaadsfadaWgaaWcbaGaaGymaiaaikdaaeqaaOGa ey4kaSIaam4uamaaBaaaleaacaaIXaGaaG4maaqabaGccaWGubWaaS baaSqaaiaaigdacaaIZaaabeaaaOqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWkca WGtbWaaSbaaSqaaiaaikdacaaIXaaabeaakiaadsfadaWgaaWcbaGa aGOmaiaaigdaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaaIYaGaaG OmaaqabaGccaWGubWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgUca RiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaamivamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4kaSIaam4uam aaBaaaleaacaaIZaGaaGymaaqabaGccaWGubWaaSbaaSqaaiaaioda caaIXaaabeaakiabgUcaRiaadofadaWgaaWcbaGaaG4maiaaikdaae qaaOGaamivamaaBaaaleaacaaIZaGaaGOmaaqabaGccqGHRaWkcaWG tbWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadsfadaWgaaWcbaGaaG 4maiaaiodaaeqaaaaaaa@89A1@

Observe that S:T=T:S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaaiOoaiaahsfacqGH9aqpca WHubGaaiOoaiaahofaaaa@38B9@ , and also that S:I=trace(S) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaaiOoaiaahMeacqGH9aqpca qG0bGaaeOCaiaabggacaqGJbGaaeyzaiaabIcacaWHtbGaaeykaaaa @3D08@ , where I is the identity tensor.

 Outer product: Let S and T be two second order tensors.  The outer product of S and T is a scalar, denoted by ST MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaeyyXICTaaCivaa aa@3912@ .  Represent S and T by their components in a basis.  Then

ST= S 11 T 11 + S 21 T 12 + S 31 T 13 + S 12 T 21 + S 22 T 22 + S 32 T 23 + S 13 T 31 + S 23 T 32 + S 33 T 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahofacqGHflY1cqGHflY1ca WHubGaeyypa0Jaam4uamaaBaaaleaacaaIXaGaaGymaaqabaGccaWG ubWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaadofadaWgaa WcbaGaaGOmaiaaigdaaeqaaOGaamivamaaBaaaleaacaaIXaGaaGOm aaqabaGccqGHRaWkcaWGtbWaaSbaaSqaaiaaiodacaaIXaaabeaaki aadsfadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abgUcaRiaadofadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaamivamaa BaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkcaWGtbWaaSbaaSqaai aaikdacaaIYaaabeaakiaadsfadaWgaaWcbaGaaGOmaiaaikdaaeqa aOGaey4kaSIaam4uamaaBaaaleaacaaIZaGaaGOmaaqabaGccaWGub WaaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRa WkcaWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaakiaadsfadaWgaaWc baGaaG4maiaaigdaaeqaaOGaey4kaSIaam4uamaaBaaaleaacaaIYa GaaG4maaqabaGccaWGubWaaSbaaSqaaiaaiodacaaIYaaabeaakiab gUcaRiaadofadaWgaaWcbaGaaG4maiaaiodaaeqaaOGaamivamaaBa aaleaacaaIZaGaaG4maaqabaaaaaa@8D77@

Observe that ST= S T :T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaeyyXICTaaCivai abg2da9iaahofadaahaaWcbeqaaiaadsfaaaGccaGG6aGaaCivaaaa @3D9F@

 

 

 Determinant

 

The determinant of a tensor is defined as the determinant of the matrix of its components in a basis.  For a second order tensor

detS=det[ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 ] = S 11 ( S 22 S 33 S 23 S 32 )+ S 12 ( S 23 S 31 S 21 S 33 )+ S 13 ( S 21 S 32 S 31 S 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiGacsgacaGGLbGaaiiDaiaaho facqGH9aqpciGGKbGaaiyzaiaacshadaWadaabaeqabaGaam4uamaa BaaaleaacaaIXaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caWGtb WaaSbaaSqaaiaaigdacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caWGtbWaaSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiaadofada WgaaWcbaGaaGOmaiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8Uaam4u amaaBaaaleaacaaIYaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7ca WGtbWaaSbaaSqaaiaaikdacaaIZaaabeaakiaaykW7aeaacaWGtbWa aSbaaSqaaiaaiodacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaado fadaWgaaWcbaGaaG4maiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8Ua am4uamaaBaaaleaacaaIZaGaaG4maaqabaaaaOGaay5waiaaw2faaa qaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeyypa0Jaam 4uamaaBaaaleaacaaIXaGaaGymaaqabaGccaGGOaGaam4uamaaBaaa leaacaaIYaGaaGOmaaqabaGccaWGtbWaaSbaaSqaaiaaiodacaaIZa aabeaakiabgkHiTiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGa am4uamaaBaaaleaacaaIZaGaaGOmaaqabaGccaGGPaGaey4kaSIaam 4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccaGGOaGaam4uamaaBaaa leaacaaIYaGaaG4maaqabaGccaWGtbWaaSbaaSqaaiaaiodacaaIXa aabeaakiabgkHiTiaadofadaWgaaWcbaGaaGOmaiaaigdaaeqaaOGa am4uamaaBaaaleaacaaIZaGaaG4maaqabaGccaGGPaGaey4kaSIaam 4uamaaBaaaleaacaaIXaGaaG4maaqabaGccaGGOaGaam4uamaaBaaa leaacaaIYaGaaGymaaqabaGccaWGtbWaaSbaaSqaaiaaiodacaaIYa aabeaakiabgkHiTiaadofadaWgaaWcbaGaaG4maiaaigdaaeqaaOGa am4uamaaBaaaleaacaaIYaGaaGOmaaqabaGccaGGPaaaaaa@B6E3@

 

Note that if S and T are two tensors, then

det(S)=det( S T )det(ST)=det(S)det(T) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC 4uaiaacMcacqGH9aqpciGGKbGaaiyzaiaacshadaqadaqaaiaahofa daahaaWcbeqaaiaadsfaaaaakiaawIcacaGLPaaacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaciizaiaacwgacaGG0bGaaiikaiaahofacq GHflY1caWHubGaaiykaiabg2da9iGacsgacaGGLbGaaiiDaiaacIca caWHtbGaaiykaiGacsgacaGGLbGaaiiDaiaacIcacaWHubGaaiykaa aa@6648@

 

 Inverse

 

Let S be a second order tensor.  The inverse of S exists if and only if det(S)0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC 4uaiaacMcacqGHGjsUcaaIWaaaaa@3A46@ , and is defined by

S 1 S=I MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbWaaWbaaSqabeaacqGHsislca aIXaaaaOGaeyyXICTaaC4uaiabg2da9iaahMeaaaa@3A7E@

where S 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahofadaahaa WcbeqaaiabgkHiTiaaigdaaaaaaa@38EA@  denotes the inverse of S and I is the identity tensor.

 

The inverse of a tensor may be computed by calculating the inverse of the matrix of its components.  The result cannot be expressed in a compact form for a general three dimensional second order tensor, and is best computed by methods such as Gaussian elimination.

 

 Eigenvalues and Eigenvectors (Principal values and direction)

 

Let S be a second order tensor.  The scalars λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@  and unit vectors m which satisfy

Sm=λm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyyXICTaaCyBaiabg2da9i abeU7aSjaah2gaaaa@3A91@

are known as the eigenvalues and eigenvectors of S, or the principal values and principal directions of S. Note that λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@  may be complex.  For a second order tensor in three dimensions, there are generally three values of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@  and three unique unit vectors m which satisfy this equation.  Occasionally, there may be only two or one value of λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@ .  If this is the case, there are infinitely many possible vectors m that satisfy the equation.  The eigenvalues of a tensor, and the components of the eigenvectors, may be computed by finding the eigenvalues and eigenvectors of the matrix of components (see A.3.2)

 

The eigenvalues of a symmetric tensor are always real.  The eigenvalues of a skew tensor are always pure imaginary or zero.

 

 Change of Basis.

 

Let S be a tensor, and 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.  Suppose that the components of S in the basis { 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@  are known to be

[ S (e) ]=[ S 11 (e) S 12 (e) S 13 (e) S 21 (e) S 22 (e) S 23 (e) S 31 (e) S 32 (e) S 33 (e) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaahwgacaGGPaaaaOGaaiyxaiabg2da9maadmaabaqbaeqabmWa aaqaaiaadofadaqhaaWcbaGaaGymaiaaigdaaeaacaGGOaGaaCyzai aacMcaaaaakeaacaWGtbWaa0baaSqaaiaaigdacaaIYaaabaGaaiik aiaahwgacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIXaGaaG4maa qaaiaacIcacaWHLbGaaiykaaaaaOqaaiaadofadaqhaaWcbaGaaGOm aiaaigdaaeaacaGGOaGaaCyzaiaacMcaaaaakeaacaWGtbWaa0baaS qaaiaaikdacaaIYaaabaGaaiikaiaahwgacaGGPaaaaaGcbaGaam4u amaaDaaaleaacaaIYaGaaG4maaqaaiaacIcacaWHLbGaaiykaaaaaO qaaiaadofadaqhaaWcbaGaaG4maiaaigdaaeaacaGGOaGaaCyzaiaa cMcaaaaakeaacaWGtbWaa0baaSqaaiaaiodacaaIYaaabaGaaiikai aahwgacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIZaGaaG4maaqa aiaacIcacaWHLbGaaiykaaaaaaaakiaawUfacaGLDbaaaaa@6628@

 

Now, suppose that we wish to compute the components of  S in a second Cartesian basis, { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0E@ .  Denote these components by

[ S (m) ]=[ S 11 (m) S 12 (m) S 13 (m) S 21 (m) S 22 (m) S 23 (m) S 31 (m) S 32 (m) S 33 (m) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaah2gacaGGPaaaaOGaaiyxaiabg2da9maadmaabaqbaeqabmWa aaqaaiaadofadaqhaaWcbaGaaGymaiaaigdaaeaacaGGOaGaaCyBai aacMcaaaaakeaacaWGtbWaa0baaSqaaiaaigdacaaIYaaabaGaaiik aiaah2gacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIXaGaaG4maa qaaiaacIcacaWHTbGaaiykaaaaaOqaaiaadofadaqhaaWcbaGaaGOm aiaaigdaaeaacaGGOaGaaCyBaiaacMcaaaaakeaacaWGtbWaa0baaS qaaiaaikdacaaIYaaabaGaaiikaiaah2gacaGGPaaaaaGcbaGaam4u amaaDaaaleaacaaIYaGaaG4maaqaaiaacIcacaWHTbGaaiykaaaaaO qaaiaadofadaqhaaWcbaGaaG4maiaaigdaaeaacaGGOaGaaCyBaiaa cMcaaaaakeaacaWGtbWaa0baaSqaaiaaiodacaaIYaaabaGaaiikai aah2gacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIZaGaaG4maaqa aiaacIcacaWHTbGaaiykaaaaaaaakiaawUfacaGLDbaaaaa@6678@

To do so, first compute the components of the transformation matrix [Q]

[ Q ]=[ m 1 e 1 m 1 e 2 m 1 e 3 m 2 e 1 m 2 e 2 m 2 e 3 m 3 e 1 m 3 e 2 m 3 e 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadgfaaiaawUfacaGLDb aacqGH9aqpdaWadaabaeqabaGaaCyBamaaBaaaleaacaaIXaaabeaa kiabgwSixlaahwgadaWgaaWcbaGaaGymaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWHTbWaaSbaaSqaaiaaigdaaeqa aOGaeyyXICTaaCyzamaaBaaaleaacaaIYaaabeaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaGymaaqa baGccqGHflY1caWHLbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaaCyBam aaBaaaleaacaaIYaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaGym aaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHTb WaaSbaaSqaaiaaikdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaI YaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2 gadaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaa iodaaeqaaaGcbaGaaCyBamaaBaaaleaacaaIZaaabeaakiabgwSixl aahwgadaWgaaWcbaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWHTbWaaSbaaSqaaiaaiodaaeqaaOGaeyyXIC TaaCyzamaaBaaaleaacaaIYaaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaG4maaqabaGccqGHfl Y1caWHLbWaaSbaaSqaaiaaiodaaeqaaaaakiaawUfacaGLDbaaaaa@A6BB@

(this is the same matrix you would use to transform vector components from { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@  to { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0E@  ).  Then,

[ S (m) ]=[Q][ S (e) ] [Q] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaah2gacaGGPaaaaOGaaiyxaiabg2da9iaacUfacaWGrbGaaiyx aiaacUfacaWGtbWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaGcca GGDbGaai4waiaadgfacaGGDbWaaWbaaSqabeaacaWGubaaaaaa@4431@

or, written out in full

[ S 11 (m) S 12 (m) S 13 (m) S 21 (m) S 22 (m) S 23 (m) S 31 (m) S 32 (m) S 33 (m) ]=[ m 1 e 1 m 1 e 2 m 1 e 3 m 2 e 1 m 2 e 2 m 2 e 3 m 3 e 1 m 3 e 2 m 3 e 3 ][ S 11 (e) S 12 (e) S 13 (e) S 21 (e) S 22 (e) S 23 (e) S 31 (e) S 32 (e) S 33 (e) ][ m 1 e 1 m 2 e 1 m 3 e 1 m 1 e 2 m 2 e 2 m 3 e 2 m 1 e 3 m 2 e 3 m 3 e 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb Waa0baaSqaaiaaigdacaaIXaaabaGaaiikaiaah2gacaGGPaaaaaGc baGaam4uamaaDaaaleaacaaIXaGaaGOmaaqaaiaacIcacaWHTbGaai ykaaaaaOqaaiaadofadaqhaaWcbaGaaGymaiaaiodaaeaacaGGOaGa aCyBaiaacMcaaaaakeaacaWGtbWaa0baaSqaaiaaikdacaaIXaaaba Gaaiikaiaah2gacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIYaGa aGOmaaqaaiaacIcacaWHTbGaaiykaaaaaOqaaiaadofadaqhaaWcba GaaGOmaiaaiodaaeaacaGGOaGaaCyBaiaacMcaaaaakeaacaWGtbWa a0baaSqaaiaaiodacaaIXaaabaGaaiikaiaah2gacaGGPaaaaaGcba Gaam4uamaaDaaaleaacaaIZaGaaGOmaaqaaiaacIcacaWHTbGaaiyk aaaaaOqaaiaadofadaqhaaWcbaGaaG4maiaaiodaaeaacaGGOaGaaC yBaiaacMcaaaaaaaGccaGLBbGaayzxaaGaeyypa0ZaamWaaqaabeqa aiaah2gadaWgaaWcbaGaaGymaaqabaGccqGHflY1caWHLbWaaSbaaS qaaiaaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaCyBamaaBaaaleaacaaIXaaabeaakiabgwSixlaahwgadaWgaa WcbaGaaGOmaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWHTbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBa aaleaacaaIZaaabeaaaOqaaiaah2gadaWgaaWcbaGaaGOmaaqabaGc cqGHflY1caWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIYaaabeaa kiabgwSixlaahwgadaWgaaWcbaGaaGOmaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWHTbWaaSbaaSqaaiaaikdaaeqa aOGaeyyXICTaaCyzamaaBaaaleaacaaIZaaabeaaaOqaaiaah2gada WgaaWcbaGaaG4maaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaaigda aeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyBam aaBaaaleaacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaGOm aaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHTb WaaSbaaSqaaiaaiodaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaI ZaaabeaaaaGccaGLBbGaayzxaaWaamWaaeaafaqabeWadaaabaGaam 4uamaaDaaaleaacaaIXaGaaGymaaqaaiaacIcacaWHLbGaaiykaaaa aOqaaiaadofadaqhaaWcbaGaaGymaiaaikdaaeaacaGGOaGaaCyzai aacMcaaaaakeaacaWGtbWaa0baaSqaaiaaigdacaaIZaaabaGaaiik aiaahwgacaGGPaaaaaGcbaGaam4uamaaDaaaleaacaaIYaGaaGymaa qaaiaacIcacaWHLbGaaiykaaaaaOqaaiaadofadaqhaaWcbaGaaGOm aiaaikdaaeaacaGGOaGaaCyzaiaacMcaaaaakeaacaWGtbWaa0baaS qaaiaaikdacaaIZaaabaGaaiikaiaahwgacaGGPaaaaaGcbaGaam4u amaaDaaaleaacaaIZaGaaGymaaqaaiaacIcacaWHLbGaaiykaaaaaO qaaiaadofadaqhaaWcbaGaaG4maiaaikdaaeaacaGGOaGaaCyzaiaa cMcaaaaakeaacaWGtbWaa0baaSqaaiaaiodacaaIZaaabaGaaiikai aahwgacaGGPaaaaaaaaOGaay5waiaaw2faamaadmaaeaqabeaacaWH TbWaaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaaca aIXaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa h2gadaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWHLbWaaSbaaSqaai aaigdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aCyBamaaBaaaleaacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcba GaaGymaaqabaaakeaacaWHTbWaaSbaaSqaaiaaigdaaeqaaOGaeyyX ICTaaCyzamaaBaaaleaacaaIYaaabeaakiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaGOmaaqabaGccqGH flY1caWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIZaaabeaakiab gwSixlaahwgadaWgaaWcbaGaaGOmaaqabaaakeaacaWHTbWaaSbaaS qaaiaaigdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaIZaaabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2gadaWgaa WcbaGaaGOmaaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaaiodaaeqa aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCyBamaaBa aaleaacaaIZaaabeaakiabgwSixlaahwgadaWgaaWcbaGaaG4maaqa baaaaOGaay5waiaaw2faaaaa@6EF1@

 

To prove this result, let u and v be vectors satisfying

v=Su MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaC4uaiabgwSixl aahwhaaaa@38EE@

Denote the components of u and v in the two bases by  u (e) _ , u (m) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadwhadaahaaWcbeqaai aacIcacaWHLbGaaiykaaaaaaGccaGGSaGaaGPaVlaaykW7caaMc8Ua aGPaVpaamaaabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPa aaaaaaaaa@40AF@  and v (e) _ , v (m) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadAhadaahaaWcbeqaai aacIcacaWHLbGaaiykaaaaaaGccaGGSaGaaGPaVlaaykW7caaMc8Ua aGPaVpaamaaabaGaamODamaaCaaaleqabaGaaiikaiaah2gacaGGPa aaaaaaaaa@40B1@ , respectively.  Recall that the vector components are related by

u (m) _ =[Q] u (e) _ u (e) _ = [Q] T u (m) _ v (m) _ =[Q] v (e) _ v (e) _ = [Q] T v (m) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaamaaabaGaamyDamaaCaaale qabaGaaiikaiaah2gacaGGPaaaaaaakiabg2da9iaacUfacaWGrbGa aiyxamaamaaabaGaamyDamaaCaaaleqabaGaaiikaiaahwgacaGGPa aaaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVp aamaaabaGaamyDamaaCaaaleqabaGaaiikaiaahwgacaGGPaaaaaaa kiabg2da9iaacUfacaWGrbGaaiyxamaaCaaaleqabaGaamivaaaakm aamaaabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPaaaaaaa aOqaamaamaaabaGaamODamaaCaaaleqabaGaaiikaiaah2gacaGGPa aaaaaakiabg2da9iaacUfacaWGrbGaaiyxamaamaaabaGaamODamaa CaaaleqabaGaaiikaiaahwgacaGGPaaaaaaakiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVpaamaaabaGaamODamaaCaaale qabaGaaiikaiaahwgacaGGPaaaaaaakiabg2da9iaacUfacaWGrbGa aiyxamaaCaaaleqabaGaamivaaaakmaamaaabaGaamODamaaCaaale qabaGaaiikaiaah2gacaGGPaaaaaaaaaaa@8E00@

Now, we could express the tensor-vector product in either basis

v (m) _ =[ S (m) ] u (m) _ v (e) _ =[ S (e) ] u (e) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadAhadaahaaWcbeqaai aacIcacaWHTbGaaiykaaaaaaGccqGH9aqpdaWadaqaaiaadofadaah aaWcbeqaaiaacIcacaWHTbGaaiykaaaaaOGaay5waiaaw2faamaama aabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPaaaaaaakiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaWaaaeaaca WG2bWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaaaaOGaeyypa0Za amWaaeaacaWGtbWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaaaki aawUfacaGLDbaadaadaaqaaiaadwhadaahaaWcbeqaaiaacIcacaWH LbGaaiykaaaaaaGccaaMc8UaaGPaVdaa@742E@

Substitute for u (e) _ , v (e) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadwhadaahaaWcbeqaai aacIcacaWHLbGaaiykaaaaaaGccaGGSaGaaGPaVlaaykW7caaMc8Ua aGPaVpaamaaabaGaamODamaaCaaaleqabaGaaiikaiaahwgacaGGPa aaaaaaaaa@40A8@  from above into the second of these two relations, we see that

[ Q ] T v (m) _ =[ S (e) ] [ Q ] T u (m) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadgfaaiaawUfacaGLDb aadaahaaWcbeqaaiaadsfaaaGcdaadaaqaaiaadAhadaahaaWcbeqa aiaacIcacaWHTbGaaiykaaaaaaGccqGH9aqpdaWadaqaaiaadofada ahaaWcbeqaaiaacIcacaWHLbGaaiykaaaaaOGaay5waiaaw2faamaa dmaabaGaamyuaaGaay5waiaaw2faamaaCaaaleqabaGaamivaaaakm aamaaabaGaamyDamaaCaaaleqabaGaaiikaiaah2gacaGGPaaaaaaa kiaaykW7aaa@496F@

Recall that

[ Q ] [ Q ] T =[ I ][ I ] v (m) _ = v (m) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiaadgfaaiaawUfacaGLDb aadaWadaqaaiaadgfaaiaawUfacaGLDbaadaahaaWcbeqaaiaadsfa aaGccqGH9aqpdaWadaqaaiaadMeaaiaawUfacaGLDbaacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8+aamWaaeaacaWGjbaacaGLBbGaayzxaaWaaWaaaeaacaWG 2bWaaWbaaSqabeaacaGGOaGaaCyBaiaacMcaaaaaaOGaeyypa0ZaaW aaaeaacaWG2bWaaWbaaSqabeaacaGGOaGaaCyBaiaacMcaaaaaaaaa @655A@

so multiplying both sides by [Q] shows that

v (m) _ =[ Q ][ S (e) ] [ Q ] T u (m) _ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaadaaqaaiaadAhadaahaaWcbeqaai aacIcacaWHTbGaaiykaaaaaaGccqGH9aqpdaWadaqaaiaadgfaaiaa wUfacaGLDbaadaWadaqaaiaadofadaahaaWcbeqaaiaacIcacaWHLb GaaiykaaaaaOGaay5waiaaw2faamaadmaabaGaamyuaaGaay5waiaa w2faamaaCaaaleqabaGaamivaaaakmaamaaabaGaamyDamaaCaaale qabaGaaiikaiaah2gacaGGPaaaaaaakiaaykW7aaa@485F@

so, comparing with the first of equation (1)

[ S (m) ]=[Q][ S (e) ] [Q] T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaCaaaleqabaGaai ikaiaah2gacaGGPaaaaOGaaiyxaiabg2da9iaacUfacaWGrbGaaiyx aiaacUfacaWGtbWaaWbaaSqabeaacaGGOaGaaCyzaiaacMcaaaGcca GGDbGaai4waiaadgfacaGGDbWaaWbaaSqabeaacaWGubaaaaaa@4431@

as stated.

 

 Invariants

 

Invariants of a tensor are functions of the tensor components which remain constant under a basis change.  That is to say, the invariant has the same value when computed in two arbitrary bases { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF6@  and { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0E@ .  A symmetric second order tensor always has three independent invariants.

 

Examples of invariants are

1.      The three eigenvalues

2.      The determinant

3.      The trace

4.      The inner and outer products

These are not all independent MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for example any of 2-4 can be calculated in terms of 1.

 

 

 

B3 SPECIAL TENSORS

 

 Identity tensor  The identity tensor I is the tensor such that, for any tensor S or vector v

Iv=vI=v SI=IS=S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahMeacqGHflY1caWH2bGaey ypa0JaaCODaiabgwSixlaahMeacqGH9aqpcaWH2baabaGaaC4uaiab gwSixlaahMeacqGH9aqpcaWHjbGaeyyXICTaaC4uaiabg2da9iaaho faaaaa@48E5@

In any basis, the identity tensor has components

[ 1 0 0 0 1 0 0 0 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaigdaaaaacaGLBbGaayzxaaaaaa@3B5B@

 

 Symmetric Tensor A symmetric tensor S has the property

S= S T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyypa0JaaC4uamaaCaaale qabaGaamivaaaaaaa@3689@

The components of a symmetric tensor have the form

[ S 11 S 12 S 13 S 12 S 22 S 23 S 13 S 23 S 33 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaWGtb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadofadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaam4uamaaBaaaleaacaaIXaGaaG4maa qabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaa dofadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaam4uamaaBaaale aacaaIYaGaaG4maaqabaaakeaacaWGtbWaaSbaaSqaaiaaigdacaaI ZaaabeaaaOqaaiaadofadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcba Gaam4uamaaBaaaleaacaaIZaGaaG4maaqabaaaaaGccaGLBbGaayzx aaaaaa@4B84@

so that there are only six independent components of the tensor, instead of nine.

 

 Skew Tensor  A skew tensor S has the property

S T =S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbWaaWbaaSqabeaacaWGubaaaO Gaeyypa0JaeyOeI0IaaC4uaaaa@3780@

The components of a skew tensor have the form

[ 0 S 12 S 13 S 12 0 S 23 S 13 S 23 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacaaIWa aabaGaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaaakeaacaWGtbWa aSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiabgkHiTiaadofadaWgaa WcbaGaaGymaiaaikdaaeqaaaGcbaGaaGimaaqaaiaadofadaWgaaWc baGaaGOmaiaaiodaaeqaaaGcbaGaeyOeI0Iaam4uamaaBaaaleaaca aIXaGaaG4maaqabaaakeaacqGHsislcaWGtbWaaSbaaSqaaiaaikda caaIZaaabeaaaOqaaiaaicdaaaaacaGLBbGaayzxaaaaaa@48E7@

 

 Orthogonal Tensors An orthogonal tensor S has the property

S S T = S T S=I S 1 = S T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahofacqGHflY1caWHtbWaaW baaSqabeaacaWGubaaaOGaeyypa0JaaC4uamaaCaaaleqabaGaamiv aaaakiabgwSixlaahofacqGH9aqpcaWHjbaabaGaaC4uamaaCaaale qabaGaeyOeI0IaaGymaaaakiabg2da9iaahofadaahaaWcbeqaaiaa dsfaaaaaaaa@4571@

An orthogonal tensor must have det(S)=±1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC 4uaiaacMcacqGH9aqpcqGHXcqScaaIXaaaaa@3B74@ ; a tensor with det(S)=+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGKbGaaiyzaiaacshacaGGOaGaaC 4uaiaacMcacqGH9aqpcqGHRaWkcaaIXaaaaa@3A68@  is known as a proper orthogonal tensor.