Page 49 - Intro to Tensor Calculus
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                        i
               Letting T ,i =1,...,N denote the components of this tangent vector in the barred system of coordinates,
               the equation (1.2.39) can then be expressed in the form

                                                  i   ∂x i  j
                                                 T =     T ,   i, j =1,... ,N.                        (1.2.40)
                                                      ∂x j
               This equation is said to define the transformation law associated with an absolute contravariant tensor of
               rank or order one. In the case N = 3 the matrix form of this transformation is represented
                                                          1    1    1  
                                                1      ∂x   ∂x   ∂x      1  
                                                T        ∂x 1  ∂x  2  ∂x 3  T
                                                 2       ∂x 2  ∂x 2  ∂x 2 
                                                T    = 
                                                                       T  2                       (1.2.41)
                                                 3       ∂x 1 3  ∂x 2 3  ∂x 3   3
                                                                     3
                                                T        ∂x   ∂x   ∂x     T
                                                         ∂x 1  ∂x 2  ∂x 3
               A more general definition is
                                                                                          i
                               Definition: (Contravariant tensor) Whenever N quantities A in
                                                                                        i
                                                    1
                                                          N
                               a coordinate system (x ,... ,x ) are related to N quantities A in a
                                                  1
                                                         N
                               coordinate system (x ,... , x ) such that the Jacobian J is different
                               from zero, then if the transformation law
                                                         i   W  ∂x i  j
                                                       A = J       A
                                                                ∂x j
                               is satisfied, these quantities are called the components of a relative tensor
                               of rank or order one with weight W. Whenever W = 0 these quantities
                               are called the components of an absolute tensor of rank or order one.

               We see that the above transformation law satisfies the group properties.


               EXAMPLE 1.2-3. (Transitive Property of Contravariant Transformation)
                   Show that successive contravariant transformations is also a contravariant transformation.
               Solution: Consider the transformation of a vector from an unbarred to a barred system of coordinates. A
                                                       i
                                                  i
               vector or absolute tensor of rank one A = A (x),i =1,... ,N will transform like the equation (1.2.40) and
                                                       i      ∂x i  j
                                                      A (x)=     A (x).                               (1.2.42)
                                                              ∂x j
               Another transformation from x → x coordinates will produce the components

                                                                i
                                                        i     ∂x   j
                                                      A (x)=    j  A (x)                              (1.2.43)
                                                              ∂x
                                               j
                                                                                                 j
               Here we have used the notation A (x) to emphasize the dependence of the components A upon the x
               coordinates. Changing indices and substituting equation (1.2.42) into (1.2.43) we find

                                                              i
                                                     i     ∂x ∂x j
                                                                    m
                                                   A (x)=          A (x).                             (1.2.44)
                                                             j
                                                           ∂x ∂x m
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