Page 51 - Intro to Tensor Calculus
P. 51

47





                               Definition: (Covariant tensor)     Whenever N quantities A i in a
                                                        N
                                                  1
                               coordinate system (x ,...,x ) are related to N quantities A i in a co-
                                               1
                                                      N
                               ordinate system (x ,..., x ), with Jacobian J different from zero, such
                               that the transformation law
                                                                ∂x j
                                                             W
                                                       A i = J    i  A j               (1.2.48)
                                                                ∂x
                               is satisfied, then these quantities are called the components of a relative
                               covariant tensor of rank or order one having a weight of W. When-
                               ever W = 0, these quantities are called the components of an absolute
                               covariant tensor of rank or order one.



                   Again we note that the above transformation satisfies the group properties. Absolute tensors of rank or
               order one are referred to as vectors while absolute tensors of rank or order zero are referred to as scalars.
                EXAMPLE 1.2-4. (Transitive Property of Covariant Transformation)
                   Consider a sequence of transformation laws of the type defined by the equation (1.2.47)

                                                                       ∂x j
                                                x → x      A i (x)= A j (x)  i
                                                                       ∂x
                                                x → x     A k (x)= A m (x) ∂x m
                                                                        ∂x k

               We can therefore express the transformation of the components associated with the coordinate transformation
               x → x and
                                                             j     m           j
                                                           ∂x    ∂x          ∂x
                                            A k (x)=  A j (x)  m   k  = A j (x)  k  ,
                                                          ∂x     ∂x         ∂x
               which demonstrates the transitive property of a covariant transformation.



               Higher Order Tensors

                   We have shown that first order tensors are quantities which obey certain transformation laws. Higher
               order tensors are defined in a similar manner and also satisfy the group properties. We assume that we are
               given transformations of the type illustrated in equations (1.2.30) and (1.2.32) which are single valued and
                                                                                     i
                                                                               i
               continuous with Jacobian J different from zero. Further, the quantities x and x ,i =1,... ,n represent the
               coordinates in any two coordinate systems. The following transformation laws define second order and third
               order tensors.
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