The coordinate transformation law for this bi-tensor is given by

 :Cr 8gb'
 CCd O Cab,. (2-29)

In addition, the usual operations such as contraction and covariant differentiations

may be immediately extended to bi-tensors with the precautions: (i) contraction

may be performed only over the indices referring to the same point, (ii) in taking

covariant derivatives all indices except those referring to the variable in question

should be ignored. One may take covariant derivatives with respect to either

variable,


 Cab; ab + CICebl, (2-30)

 Ca b'd' b',d' Ffi /d- C (2-31)


where the semicolon denotes covariant differentiation and the comma denotes

ordinary differentiation. Indices associated with covariant differentiation at different

points commute, while the usual commutation laws hold for indices referring to the

same point.

 One may define a bi-scalar which is an invariant bi-tensor bearing no indices.

One may also introduce a u.:-.1 ,h.:'/; and its most elementary example is the

four-dimensional delta function


 6() (x,) 6(x0 )6(x1 1)6(x2 2)6(x3 Z3) 6()(,x). (2-32)


In general, the delta function may be regarded as a density of weight w at the

point x and weight 1 w at the point z, where w is arbitrary. One may choose

w = 1/2 for the sake of symmetry, and the transformation law for the delta

function may be give in the form

 (4) 1/2 z 1/2
 ax(1" )a -.46 (x,z). (2-33)