Appendix E

 

Miscellaneous derivations

 

 

E.1. Relation between the areas of the faces of a tetrahedron

 

For the tetrahedron shown, we prove that:

d A 1 d A (n) = n 1 d A 2 d A (n) = n 2 d A 3 d A (n) = n 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadgeadaWgaaWcba GaaGymaaqabaaakeaacaWGKbGaamyqamaaBaaaleaacaGGOaGaamOB aiaacMcaaeqaaaaakiabg2da9iaad6gadaWgaaWcbaGaaGymaaqaba GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8+aaSaaaeaacaWGKbGaamyqamaaBaaaleaacaaIYa aabeaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaacIcacaWGUbGaaiyk aaqabaaaaOGaeyypa0JaamOBamaaBaaaleaacaaIYaaabeaakiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VpaalaaabaGaamizaiaadgeadaWgaaWcbaGaaG4maaqabaaakeaaca WGKbGaamyqamaaBaaaleaacaGGOaGaamOBaiaacMcaaeqaaaaakiab g2da9iaad6gadaWgaaWcbaGaaG4maaqabaGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8oaaa@75EC@

where d A (n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaacIcaca WGUbGaaiykaaqabaaaaa@348E@  is the area of the face with normal n, and d A i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaadMgaae qaaaaa@3330@  is the face with normal e i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaahwgadaWgaaWcbaGaamyAaa qabaaaaa@335C@

 

Note that

d A 1 = 1 2 bcd A 2 = 1 2 acd A 3 = 1 2 ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaaigdaae qaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGIbGaam4y aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaamizaiaadgeadaWgaaWcbaGaaGOmaaqa baGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadggacaWGJb GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadsgacaWGbbWaaSbaaSqaaiaaiodaaeqaaOGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGHbGaamOyaaaa@65C2@

Note also that we can compute the area of the face with normal n by taking cross products of the vectors defining the sides of the face:

d A n n= 1 2 ( c e 3 b e 2 )×( a e 1 b e 2 )= 1 2 ( bc e 1 +ac e 2 +ab e 3 ) d A n n=d A 1 e 1 +d A 2 e 2 +d A 3 e 3 n= d A 1 d A n e 1 + d A 2 d A n e 2 + d A 3 d A n e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamizaiaadgeadaWgaaWcbaGaam OBaaqabaGccaWHUbGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaa daqadaqaaiaadogacaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0 IaamOyaiaahwgadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaa cqGHxdaTdaqadaqaaiaadggacaWHLbWaaSbaaSqaaiaaigdaaeqaaO GaeyOeI0IaamOyaiaahwgadaWgaaWcbaGaaGOmaaqabaaakiaawIca caGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaaba GaamOyaiaadogacaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIa amyyaiaadogacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaam yyaiaadkgacaWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzk aaaabaGaeyO0H4TaamizaiaadgeadaWgaaWcbaGaamOBaaqabaGcca WHUbGaeyypa0JaamizaiaadgeadaWgaaWcbaGaaGymaaqabaGccaWH LbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamizaiaadgeadaWgaa WcbaGaaGOmaaqabaGccaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaey4k aSIaamizaiaadgeadaWgaaWcbaGaaG4maaqabaGccaWHLbWaaSbaaS qaaiaaiodaaeqaaaGcbaGaeyO0H4TaaCOBaiabg2da9maalaaabaGa amizaiaadgeadaWgaaWcbaGaaGymaaqabaaakeaacaWGKbGaamyqam aaBaaaleaacaWGUbaabeaaaaGccaWHLbWaaSbaaSqaaiaaigdaaeqa aOGaey4kaSYaaSaaaeaacaWGKbGaamyqamaaBaaaleaacaaIYaaabe aaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaad6gaaeqaaaaakiaahwga daWgaaWcbaGaaGOmaaqabaGccqGHRaWkdaWcaaqaaiaadsgacaWGbb WaaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaadgeadaWgaaWcbaGa amOBaaqabaaaaOGaaCyzamaaBaaaleaacaaIZaaabeaaaaaa@8C96@

so that

d A 1 d A (n) = n 1 d A 2 d A (n) = n 2 d A 3 d A (n) = n 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadgeadaWgaaWcba GaaGymaaqabaaakeaacaWGKbGaamyqamaaBaaaleaacaGGOaGaamOB aiaacMcaaeqaaaaakiabg2da9iaad6gadaWgaaWcbaGaaGymaaqaba GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8+aaSaaaeaacaWGKbGaamyqamaaBaaaleaacaaIYa aabeaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaacIcacaWGUbGaaiyk aaqabaaaaOGaeyypa0JaamOBamaaBaaaleaacaaIYaaabeaakiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VpaalaaabaGaamizaiaadgeadaWgaaWcbaGaaG4maaqabaaakeaaca WGKbGaamyqamaaBaaaleaacaGGOaGaamOBaiaacMcaaeqaaaaakiab g2da9iaad6gadaWgaaWcbaGaaG4maaqabaGccaaMc8oaaa@6CAA@

as required.

 

 

E.2. Relation between area elements before and after deformation

 

Consider an element of area d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaaicdaae qaaaaa@330C@  with normal n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah6gadaahaaWcbeqaaiaaicdaaaaaaa@3255@  in a deformable solid.  Suppose the solid is deformed, and let F ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@334B@  denote the components of the deformation gradient tensor. The area element deforms with the solid, and has a new area dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbaaaa@3226@  and normal n.  We plan to prove that the deformed area element is related to its undeformed area through

dA n i =J F ki 1 n k 0 d A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaDaaaleaaca WGPbaabaaaaOGaeyypa0JaamOsaiaadAeadaqhaaWcbaGaam4Aaiaa dMgaaeaacqGHsislcaaIXaaaaOGaamOBamaaDaaaleaacaWGRbaaba GaaGimaaaakiaadsgacaWGbbWaaSbaaSqaaiaaicdaaeqaaaaa@3FF4@

where J=det( F ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacqGH9aqpciGGKbGaaiyzaiaacs hacaGGOaGaamOramaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaa aa@393E@

 

Start by noting that the area before deformation can be computed by taking the cross product of two infinitesimal vectors d v i ,d w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG2bWaaSbaaSqaaiaadMgaae qaaOGaaiilaiaaykW7caaMc8UaaGPaVlaadsgacaWG3bWaaSbaaSqa aiaadMgaaeqaaaaa@3BBF@  bounding the area element in the undeformed configuration

d A 0 n i 0 = ijk d v j d w k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbWaaSbaaSqaaiaaicdaae qaaOGaamOBamaaDaaaleaacaWGPbaabaGaaGimaaaakiabg2da9iab gIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaamizaiaadA hadaWgaaWcbaGaamOAaaqabaGccaWGKbGaam4DamaaBaaaleaacaWG Rbaabeaaaaa@416F@

Note that the infinitesimal vectors map to F ij d w j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaamizaiaadEhadaWgaaWcbaGaamOAaaqabaaaaa@3646@ , F ij d v j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaamizaiaadAhadaWgaaWcbaGaamOAaaqabaaaaa@3645@  in the deformed configuration.  Therefore

dA n i = ijk F jm d v m F kn d w n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaDaaaleaaca WGPbaabaaaaOGaeyypa0JaeyicI48aaSbaaSqaaiaadMgacaWGQbGa am4AaaqabaGccaWGgbWaaSbaaSqaaiaadQgacaWGTbaabeaakiaads gacaWG2bWaaSbaaSqaaiaad2gaaeqaaOGaamOramaaBaaaleaacaWG RbGaamOBaaqabaGccaWGKbGaam4DamaaBaaaleaacaWGUbaabeaaaa a@4591@

Let F ij 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaqhaaWcbaGaamyAaiaadQgaae aacqGHsislcaaIXaaaaaaa@34E4@  denote the inverse of the deformation gradient tensor, i.e. F ik F kj 1 = δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadUgaae qaaOGaamOramaaDaaaleaacaWGRbGaamOAaaqaaiabgkHiTiaaigda aaGccqGH9aqpcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@3C83@ . Then, we could write

dA n i = F pl F li 1 pjk F jm F kn d v m d w n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaDaaaleaaca WGPbaabaaaaOGaeyypa0JaamOramaaBaaaleaacaWGWbGaamiBaaqa baGccaWGgbWaa0baaSqaaiaadYgacaWGPbaabaGaeyOeI0IaaGymaa aakiabgIGiopaaBaaaleaacaWGWbGaamOAaiaadUgaaeqaaOGaamOr amaaBaaaleaacaWGQbGaamyBaaqabaGccaWGgbWaaSbaaSqaaiaadU gacaWGUbaabeaakiaadsgacaWG2bWaaSbaaSqaaiaad2gaaeqaaOGa amizaiaadEhadaWgaaWcbaGaamOBaaqabaaaaa@4D08@

Now, recall the identity

λ=detAλ= 1 6 ijk lmn A li A mj A nk lmn λ= ijk A il A jm A kn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4UdWMaeyypa0Jaciizaiaacw gacaGG0bGaaCyqaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlabggMi6kaaykW7caaMc8UaaGPaVlaayk W7caaMc8Uaeq4UdWMaeyypa0JaaGPaVlaaykW7daWcaaqaaiaaigda aeaacaaI2aaaaiabgIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaae qaaOGaeyicI48aaSbaaSqaaiaadYgacaWGTbGaamOBaaqabaGccaWG bbWaaSbaaSqaaiaadYgacaWGPbaabeaakiaadgeadaWgaaWcbaGaam yBaiaadQgaaeqaaOGaamyqamaaBaaaleaacaWGUbGaam4AaaqabaGc caaMc8oabaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlabgsDiBlaaykW7caaMc8UaaGPaVlabgIGiopaaBaaaleaacaWGSb GaamyBaiaad6gaaeqaaOGaeq4UdWMaaGPaVlaaykW7cqGH9aqpcaaM c8UaaGPaVlaaykW7cqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRb aabeaakiaadgeadaWgaaWcbaGaamyAaiaadYgaaeqaaOGaamyqamaa BaaaleaacaWGQbGaamyBaaqabaGccaWGbbWaaSbaaSqaaiaadUgaca WGUbaabeaaaaaa@BB9B@

so that

dA n i = F li 1 J lmn d v m d w n = F li 1 Jd A 0 n l 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGbbGaamOBamaaDaaaleaaca WGPbaabaaaaOGaeyypa0JaamOramaaDaaaleaacaWGSbGaamyAaaqa aiabgkHiTiaaigdaaaGccaWGkbGaeyicI48aaSbaaSqaaiaadYgaca WGTbGaamOBaaqabaGccaWGKbGaamODamaaBaaaleaacaWGTbaabeaa kiaadsgacaWG3bWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamOram aaDaaaleaacaWGSbGaamyAaaqaaiabgkHiTiaaigdaaaGccaWGkbGa amizaiaadgeadaWgaaWcbaGaaGPaVlaaicdaaeqaaOGaamOBamaaDa aaleaacaWGSbaabaGaaGimaaaaaaa@5289@

where J=det( F ij ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacqGH9aqpciGGKbGaaiyzaiaacs hacaGGOaGaamOramaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaaa aa@393E@ , giving the required result.

 

 

E.3. Time derivatives of integrals over volumes within a deforming solid

 

Consider a deformable solid.  Let ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@330D@  denote the mass density of the solid in the original configuration, and let ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYbaa@37F9@  denote the mass density in the deformed configuration. (Both ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@330D@  and ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYbaa@37F9@  can vary with position in the solid). Let  V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaWgaaWcbaGaaGimaaqabaaaaa@3228@  denote a closed region within the undeformed solid, and let V be the same region of the solid in the deformed configuration.  Suppose that

v i = u i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqabaaa keaacqGHciITcaWG0baaaaaa@398F@

denotes the velocity field within the body.  We shall show that

d dt { V ρ v i dV }= V ρ d v i dt dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaacmaabaWaa8quaeaacqaHbpGCcaaMc8UaamODamaaBaaaleaa caWGPbaabeaaaeaacaWGwbaabeqdcqGHRiI8aOGaamizaiaadAfaai aawUhacaGL9baacqGH9aqpdaWdrbqaaiabeg8aYnaalaaabaGaamiz aiaadAhadaWgaaWcbaGaamyAaaqabaaakeaacaWGKbGaamiDaaaaaS qaaiaadAfaaeqaniabgUIiYdGccaWGKbGaamOvaaaa@4C6E@

and also that

d dt { V ijk y j ρ v k dV }= V ρ ijk y j d v k dt dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaacmaabaWaa8quaeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQga caWGRbaabeaakiaadMhadaWgaaWcbaGaamOAaaqabaGccqaHbpGCca aMc8UaamODamaaBaaaleaacaWGRbaabeaaaeaacaWGwbaabeqdcqGH RiI8aOGaamizaiaadAfaaiaawUhacaGL9baacqGH9aqpdaWdrbqaai abeg8aYjabgIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGa amyEamaaBaaaleaacaWGQbaabeaakmaalaaabaGaamizaiaadAhada WgaaWcbaGaam4AaaqabaaakeaacaWGKbGaamiDaaaaaSqaaiaadAfa aeqaniabgUIiYdGccaWGKbGaamOvaiaaykW7aaa@5B51@

At first glance, this looks obvious MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  just take the derivative under the integral sign.  You can’t do this, however, because the volume V changes with time, as the solid is deforming.  In addition, the mass density varies with time, because of the deformation, so even if we could take the time derivative under the integral, we’d end up with an additional term. To do the derivative properly, we first need to change variables so the integral is evaluated over the undeformed volume (which is independent of time). Thus

V ρ v i dV= V 0 ρ v i Jd V 0 = V 0 ρ 0 v i d V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeqyWdiNaamODamaaBaaale aacaWGPbaabeaakiaadsgacaWGwbGaeyypa0Zaa8quaeaacqaHbpGC caWG2bWaaSbaaSqaaiaadMgaaeqaaOGaamOsaiaadsgacaWGwbWaaS baaSqaaiaaicdaaeqaaOGaeyypa0Zaa8quaeaacqaHbpGCdaWgaaWc baGaaGimaaqabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadA fadaWgaaadbaGaaGimaaqabaaaleqaniabgUIiYdaaleaacaWGwbWa aSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aaWcbaGaamOvaaqab0 Gaey4kIipakiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaaaa@5239@

where

J=det( F ij )= dV d V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacqGH9aqpciGGKbGaaiyzaiaacs hacaGGOaGaamOramaaBaaaleaacaWGPbGaamOAaaqabaGccaGGPaGa eyypa0ZaaSaaaeaacaWGKbGaamOvaaqaaiaadsgacaWGwbWaaSbaaS qaaiaaicdaaeqaaaaaaaa@3EC2@

and we have recalled a result from the Kinematics section

Jρ= ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeacqaHbpGCcqGH9aqpcqaHbpGCda WgaaWcbaGaaGimaaqabaaaaa@36A2@

Now, we can happily differentiate.  The mass density in the undeformed configuration does not vary with time, so that

d dt { V 0 ρ 0 v i d V 0 }= V 0 ρ 0 d v i dt d V 0 = V ρ d v i dt dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaacmaabaWaa8quaeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGc caaMc8UaamODamaaBaaaleaacaWGPbaabeaaaeaacaWGwbWaaSbaaW qaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGaamizaiaadAfadaWgaaWc baGaaGimaaqabaaakiaawUhacaGL9baacqGH9aqpdaWdrbqaaiabeg 8aYnaaBaaaleaacaaIWaaabeaakmaalaaabaGaamizaiaadAhadaWg aaWcbaGaamyAaaqabaaakeaacaWGKbGaamiDaaaaaSqaaiaadAfada WgaaadbaGaaGimaaqabaaaleqaniabgUIiYdGccaWGKbGaamOvamaa BaaaleaacaaIWaaabeaakiabg2da9maapefabaGaeqyWdihaleaaca WGwbaabeqdcqGHRiI8aOWaaSaaaeaacaWGKbGaamODamaaBaaaleaa caWGPbaabeaaaOqaaiaadsgacaWG0baaaiaadsgacaWGwbaaaa@5EC5@

The last expression was obtained by changing variables in the integral back to the deformed configuration.  This is the first result we wanted.

 

To show the second result, follow exactly the same procedure, until you obtain

d dt { V 0 ijk y j v k ρ 0 d V 0 }= V 0 ρ 0 ijk ( d y j dt v k + y j d v k dt ) d V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaacmaabaWaa8quaeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQga caWGRbaabeaakiaadMhadaWgaaWcbaGaamOAaaqabaGccaaMc8Uaam ODamaaBaaaleaacaWGRbaabeaaaeaacaWGwbWaaSbaaWqaaiaaicda aeqaaaWcbeqdcqGHRiI8aOGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaO GaaGPaVlaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaaGccaGL7bGa ayzFaaGaeyypa0Zaa8quaeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba GccqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaakmaabmaa baWaaSaaaeaacaWGKbGaamyEamaaBaaaleaacaWGQbaabeaaaOqaai aadsgacaWG0baaaiaadAhadaWgaaWcbaGaam4AaaqabaGccqGHRaWk caWG5bWaaSbaaSqaaiaadQgaaeqaaOWaaSaaaeaacaWGKbGaamODam aaBaaaleaacaWGRbaabeaaaOqaaiaadsgacaWG0baaaaGaayjkaiaa wMcaaaWcbaGaamOvamaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIi pakiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaaaa@6A75@

Now, observe that

ijk d y j dt v k = ijk v j v k =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgIGiopaaBaaaleaacaWGPbGaamOAai aadUgaaeqaaOWaaSaaaeaacaWGKbGaamyEamaaBaaaleaacaWGQbaa beaaaOqaaiaadsgacaWG0baaaiaadAhadaWgaaWcbaGaam4Aaaqaba GccqGH9aqpcqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRbaabeaa kiaadAhadaWgaaWcbaGaamOAaaqabaGccaWG2bWaaSbaaSqaaiaadU gaaeqaaOGaeyypa0JaaGimaaaa@479B@

(the cross product of two parallel vectors is zero) so substituting this into the preceding equation and changing variables in the integral as before gives the required result.

 

 

 

 

E.4. Time Derivatives of the Curvature Vector for Deforming Rods

 

Consider a deforming rod, as shown in the figure.

* Let x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@  denote the position vector of a material particle on the axis of the undeformed rod;

* Let s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbaaaa@33BE@  denote the arc-length coordinate of this particle after deformation.

* Define basis vectors { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  attached to the deformed rod, following the convention described in Section 10.2.

*  Define an angular velocity vector ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHjpaaaa@341B@  and curvature vector κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH6oaaaa@340C@  through

d m i dt =ω× m i d m i ds =κ× m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHTbWaaSbaaS qaaiaadMgaaeqaaaGcbaGaamizaiaadshaaaGaeyypa0JaaCyYdiab gEna0kaah2gadaWgaaWcbaGaamyAaaqabaGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8+aaSaaaeaacaWGKbGaaCyBamaaBaaaleaacaWGPbaabeaaaO qaaiaadsgacaWGZbaaaiabg2da9iaahQ7acqGHxdaTcaWHTbWaaSba aSqaaiaadMgaaeqaaaaa@74E2@

 We shall show that the gradient of the angular velocity vector characterizing the rotation of the rod’s cross-section is related to the time derivative of the curvature vector by

dω ds = κ = dκ dt ω×κ+ d x 3 ds d s ˙ d x 3 κ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaahM8aaeaacaWGKb Gaam4CaaaacqGH9aqpdaWfGaqaaiaahQ7aaSqabeaacqGHhis0aaGc cqGH9aqpdaWcaaqaaiaadsgacaWH6oaabaGaamizaiaadshaaaGaey OeI0IaaCyYdiabgEna0kaahQ7acqGHRaWkdaWcaaqaaiaadsgacaWG 4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaadohacaWLa8oaam aalaaabaGaamizaiqadohagaGaaaqaaiaadsgacaWG4bWaaSbaaSqa aiaaiodaaeqaaaaakiaahQ7aaaa@50EE@

To see this, start by differentiating the definitions of the angular velocity vector and the curvature vector

d 2 m i dsdt = dω ds × m i +ω× d m i ds = dω ds × m i +ω×(κ× m i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgadaahaaWcbeqaai aaikdaaaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaamizaiaa dohacaWGKbGaamiDaaaacqGH9aqpdaWcaaqaaiaadsgacaWHjpaaba GaamizaiaadohaaaGaey41aqRaaCyBamaaBaaaleaacaWGPbaabeaa kiabgUcaRiaahM8acqGHxdaTdaWcaaqaaiaadsgacaWHTbWaaSbaaS qaaiaadMgaaeqaaaGcbaGaamizaiaadohaaaGaeyypa0ZaaSaaaeaa caWGKbGaaCyYdaqaaiaadsgacaWGZbaaaiabgEna0kaah2gadaWgaa WcbaGaamyAaaqabaGccqGHRaWkcaWHjpGaey41aqRaaiikaiaahQ7a cqGHxdaTcaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@61BB@

and, similarly,

d dt d m i ds = d dt ( d x 3 ds d m i d x 3 )= d s ˙ d x 3 d x 3 ds d m i ds + d 2 m i dsdt = dκ dt × m i +κ×(ω× m i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam iDaaaadaWcaaqaaiaadsgacaWHTbWaaSbaaSqaaiaadMgaaeqaaaGc baGaamizaiaadohaaaGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizai aadshaaaWaaeWaaeaadaWcaaqaaiaadsgacaWG4bWaaSbaaSqaaiaa iodaaeqaaaGcbaGaamizaiaadohaaaWaaSaaaeaacaWGKbGaaCyBam aaBaaaleaacaWGPbaabeaaaOqaaiaadsgacaWG4bWaaSbaaSqaaiaa iodaaeqaaaaaaOGaayjkaiaawMcaaiabg2da9iabgkHiTmaalaaaba GaamizaiqadohagaGaaaqaaiaadsgacaWG4bWaaSbaaSqaaiaaioda aeqaaaaakmaalaaabaGaamizaiaadIhadaWgaaWcbaGaaG4maaqaba aakeaacaWGKbGaam4CaaaadaWcaaqaaiaadsgacaWHTbWaaSbaaSqa aiaadMgaaeqaaaGcbaGaamizaiaadohaaaGaey4kaSYaaSaaaeaaca WGKbWaaWbaaSqabeaacaaIYaaaaOGaaCyBamaaBaaaleaacaWGPbaa beaaaOqaaiaadsgacaWGZbGaamizaiaadshaaaGaeyypa0ZaaSaaae aacaWGKbGaaCOUdaqaaiaadsgacaWG0baaaiabgEna0kaah2gadaWg aaWcbaGaamyAaaqabaGccqGHRaWkcaWH6oGaey41aqRaaiikaiaahM 8acqGHxdaTcaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@785A@

where s ˙ =(ds/dt) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGZbGbaiaacqGH9aqpcaGGOaGaam izaiaadohacaGGVaGaamizaiaadshacaGGPaaaaa@3A9C@  with x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@  held fixed. The preceding two results show that

dω ds × m i +ω×(κ× m i )= dκ dt × m i +κ×(ω× m i )+ d s ˙ d x 3 d x 3 ds κ× m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHjpaabaGaam izaiaadohaaaGaey41aqRaaCyBamaaBaaaleaacaWGPbaabeaakiab gUcaRiaahM8acqGHxdaTcaGGOaGaaCOUdiabgEna0kaah2gadaWgaa WcbaGaamyAaaqabaGccaGGPaGaeyypa0ZaaSaaaeaacaWGKbGaaCOU daqaaiaadsgacaWG0baaaiabgEna0kaah2gadaWgaaWcbaGaamyAaa qabaGccqGHRaWkcaWH6oGaey41aqRaaiikaiaahM8acqGHxdaTcaWH TbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiabgUcaRmaalaaabaGaam izaiqadohagaGaaaqaaiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqa aaaakmaalaaabaGaamizaiaadIhadaWgaaWcbaGaaG4maaqabaaake aacaWGKbGaam4CaaaacaWH6oGaey41aqRaaCyBamaaBaaaleaacaWG Pbaabeaaaaa@6AA6@

Next, note that we can expand the triple cross-products (see Appendix A) as

κ×(ω× m i )ω×(κ× m i )=( κ m i )ω( ωκ ) m i [ ( ω m i )κ( ωκ ) m i ] =( κ m i )ω( ω m i )κ= m i ×( κ×ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahQ7acqGHxdaTcaGGOaGaaC yYdiabgEna0kaah2gadaWgaaWcbaGaamyAaaqabaGccaGGPaGaeyOe I0IaaCyYdiabgEna0kaacIcacaWH6oGaey41aqRaaCyBamaaBaaale aacaWGPbaabeaakiaacMcacqGH9aqpdaqadaqaaiaahQ7acqGHflY1 caWHTbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaaCyYdi abgkHiTmaabmaabaGaaCyYdiabgwSixlaahQ7aaiaawIcacaGLPaaa caWHTbWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0YaamWaaeaadaqada qaaiaahM8acqGHflY1caWHTbWaaSbaaSqaaiaadMgaaeqaaaGccaGL OaGaayzkaaGaaCOUdiabgkHiTmaabmaabaGaaCyYdiabgwSixlaahQ 7aaiaawIcacaGLPaaacaWHTbWaaSbaaSqaaiaadMgaaeqaaaGccaGL BbGaayzxaaaabaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH 9aqpdaqadaqaaiaahQ7acqGHflY1caWHTbWaaSbaaSqaaiaadMgaae qaaaGccaGLOaGaayzkaaGaaCyYdiabgkHiTmaabmaabaGaaCyYdiab gwSixlaah2gadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaca WH6oGaeyypa0JaeyOeI0IaaCyBamaaBaaaleaacaWGPbaabeaakiab gEna0oaabmaabaGaaCOUdiabgEna0kaahM8aaiaawIcacaGLPaaaaa aa@EF1D@

Hence, we conclude that

dω ds × m i = dκ dt × m i +(κ×ω)× m i + d s ˙ d x 3 d x 3 ds κ× m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHjpaabaGaam izaiaadohaaaGaey41aqRaaCyBamaaBaaaleaacaWGPbaabeaakiab g2da9maalaaabaGaamizaiaahQ7aaeaacaWGKbGaamiDaaaacqGHxd aTcaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaaiikaiaahQ7a cqGHxdaTcaWHjpGaaiykaiabgEna0kaah2gadaWgaaWcbaGaamyAaa qabaGccqGHRaWkdaWcaaqaaiaadsgaceWGZbGbaiaaaeaacaWGKbGa amiEamaaBaaaleaacaaIZaaabeaaaaGcdaWcaaqaaiaadsgacaWG4b WaaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaadohaaaGaaCOUdiab gEna0kaah2gadaWgaaWcbaGaamyAaaqabaaaaa@5F88@

This result must hold for all three vectors { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ , and therefore

dω ds = dκ dt (ω×κ)+ d s ˙ d x 3 d x 3 ds κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHjpaabaGaam izaiaadohaaaGaeyypa0ZaaSaaaeaacaWGKbGaaCOUdaqaaiaadsga caWG0baaaiabgkHiTiaacIcacaWHjpGaey41aqRaaCOUdiaacMcacq GHRaWkdaWcaaqaaiaadsgaceWGZbGbaiaaaeaacaWGKbGaamiEamaa BaaaleaacaaIZaaabeaaaaGcdaWcaaqaaiaadsgacaWG4bWaaSbaaS qaaiaaiodaaeqaaaGcbaGaamizaiaadohaaaGaaCOUdaaa@4ED9@

as stated.