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)