Appendix
               Elements of Tensor Calculus
               A. Vector and Tensor Designations
               The following tensor designations are used in the book:
               a tensor of zero rank (scalar),
               a ðakÞ tensor of first rank (vector),
               AðAkjÞ tensor of second rank,
               UðdkjÞ unit tensor (dkj — Kronecker symbol),
               J ðJijkÞ tensor of third rank.
               Simmetric and Antisymmetric Tensors
               Simmetric and antisymmetric tensors are defined as follows:
               simmetric:
                                               A¼Atransp ðA ¼ A Þ                                 (A.1)
                                                               kj    jk
               antisymmetric:
                                             A¼Atransp ðA ¼A Þ                                  (A.2)
                                                               kj      jk
                  Trace of tensor is defined as the sum of its diagonal elements:
                                                   SpA¼XAkk                                       (A.3)
                                                             k
               M.Y. Marov and A.V. Kolesnichenko, Turbulence and Self-Organization:                  607
               Modeling Astrophysical Objects, Astrophysics and Space Science Library 389,
               DOI10.1007/978-1-4614-5155-6, # Springer Science+Business Media New York 2013
              608                                                          Elements of Tensor Calculus
              Scalar and Tensor (Internal) Product
              Scalar product:
                               of two vectors          ab¼Pakbk            ðÞscalar
                                                                k
                               of vector and tensor    Ab¼PAjkbk           ðÞvector
                                                                k
                               of tensor and vector    bA¼PbkAkj           ðÞvector           (A.4)
                                                                k
                               of two tensors          AB¼PAjkBki ðÞtensor
                                                                 k
              Dual scalar product oftensors:
                                           A:B¼XAjkBkj ðscalarÞ                                (A.5)
                                                     k;j
              Internal (dyad) tensor product:
                    of two vectors          ðabÞjk ¼ ajbk       ðÞtensor of second rank
                    of vector and tensor    ðaBÞijk ¼ aiBjk     ðÞtensor of third rank
                                                                                               (A.6)
                    of tensor and vector    ðBaÞijk ¼ Bijak     ðÞtensor of third rank
                    of two tensors          ðABÞijkl ¼ AijBkl   ðÞtensor of fourth rank :
              Vector product of twovectors and tensor andvector:
                    ða bÞk ¼ Xeijkaibj ðvectorÞ;       ðBaÞik ¼ XejklBijal ðtensorÞ          (A.7)
                                 i;j                                  j;l
              where symbol of permutation eijk takes the values:
                         8þ1under even permutation of indexes ði:e: 123; 231; 312Þ
                         >
                  eijk ¼ <1 under odd permutationof indexes ði:e: 321; 132; 213Þ              (A.8)
                         >
                         :0underrecurring indexes:
              B. Cylindrical Coordinates
              Expressions for the different operators used in the equations of heterogeneous
              mechanics are represented here, for the convenience, in the cylindrical coordinate
              system r;’;z (in axisymmetric case, @=@’ ¼ 0). Here are the expressions of
              operators acting on:
                       Elements of Tensor Calculus                                                                                                            609
                       (1) scalars:
                                dB@BþurB¼@Bþu @Bþu @B; rB¼i @Bþi @B                                                                                      (B.1)
                                 dt         @t                          @t         r @r           z @z                       r @r          z @z
                                                                                          
                                                                                  1 @           @B           @2B
                                                                    r2B¼                     r           þ           ;                                     (B.2)
                                                                                  r @r          @r            @z2
                       (2) vectors:
                                                                       rA¼1@ðrArÞþ@Az;                                                                    (B.3)
                                                                                      r      @r             @z
                                          @A               @A              @A              A               A            @A               @A              @A
                          rA¼ii                r þi i          ’ þi i           z þ i i       ’ þi i         r þ i i         r þ i i         ’ þi i           z ;
                                      r r @r          r ’ @r           r z @r          ’ r r         ’ ’ r          z r @z          z ’ @z           z z @z
                                                                                                                                                           (B.4)
                       (3) dyads:
                          P ¼i i P þi i P                þi i P       þi i P        þi i P         þi i P        þi i P      þii P þiiP ; (B.5)
                                  r r   rr     r ’ r’        r z   rz     ’ r ’r        ’ ’ ’’         ’ z ’z        z r  zr      z ’ z’       z z   zz
                                                   
                                                     1 @ðrP Þ              @P           P’’
                                 rP¼i                           rr            zr
                                                 r   r       @r              @z            r
                                                      
                                                         1 @ðrP Þ              @P            P                 1 @ðrP Þ              @P
                                               þi                    r’ þ           z’ þ ’r þi                             rz þ           zz :             (B.6)
                                                    ’ r          @r              @z            r           z   r       @r              @z
                            Then for strain tensor and strain velocity tensor we obtain, respectively:
                                                      1
                                              D ruþðruÞtransp
                                                      2                          
                                                  ¼i i @ur þi i 1 @u’ u’ þi i 1 @uz þ@ur
                                                        r r @r           r ’ 2       @r          r            r z 2      @r         @z
                                                               
                                                  þi i 1 @u’u’ þi i ur þi i 1 @u’
                                                       ’ r 2        @r          r           ’ ’ r           ’ z 2 @z
                                                               
                                                  þii 1 @uzþ@ur þi i 1 @u’þi i @uz;                                                                        (B.7)
                                                       z r 2       @r         @z            z ’ 2 @z              z z @z
                                         
                              D1 ruþðruÞtransp 1Iru
                                       2                                      3
                                            
                                  ¼i i         @ur 1Iru þi i 1 @u’u’ þi i 1 @uzþ@ur
                                        r r     @r        3                      r ’ 2       @r          r            r z 2      @r         @z
                                                 (B.8)
                                  þi i 1 @u’ u’ þi i                                ur 1Iru þi i 1 @u’
                                       ’ r 2        @r          r           ’ ’       r      3                      ’ z 2 @z
                                                
                                  þi i 1 @uz þ@ur þi i 1 @u’ þi i                                        @uz 1Iru :
                                       z r 2       @r         @z            z ’ 2 @z              z z     @z        3
                     610                                                                                   Elements of Tensor Calculus
                         Double-point Gibbs multiplication serves as operator widely used in hydrody-
                     namics. If, following the Gibbs notations,a;b;c;dare arbitrary vectors, thenab : cd
                     ¼ðacÞðbdÞ. In particular, for unit vectors we have:
                                                        i i  : i i   ¼i iÞði i Þ¼d d ;                                                 (B.9)
                                                         j k    l m       j    l     k    m         jl km
                     and for two dyads we have:
                                                  ! !
                                ð1Þ      ð2Þ        XX ð1Þ XX ð2Þ
                              D :D ¼                            ijikD        :               ilimD
                                                                       jk                            lm
                                                      j     k                       l    m
                                             ¼XXXXdd Dð1ÞDð2Þ¼XXDð1ÞDð2Þ;                                                              (B.10)
                                                                           jl km jk        lm                     jk     jk
                                                   j     k     l     m                                j     k
                     or
                                              2          2          2          2           2           2      2            2
                             2D:D¼2D þ2D þD þ4D þ4D þ4D  ðruÞ :                                                                      (B.11)
                                                rr         ’’         zz         r’          rz          z’     3