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Sec.
2.13
TENSOR
EQUATIONS
49
which can be reduced to
(4)
But this states that the functions
d[A/dxp
+
r'2p[s
are the components of
a mixed tensor of rank two. Hence, the
functions
are the components
of
a mixed tensor field of rank two, called the covariant
derivative
the contravariant vector
9.
We shall use
the notation
tila
for
the covariant derivative of
t'.
By a slight variation in the derivation, it can be shown that the
are the components of a covariant tensor field
rank two whenever
.$
are
the components
a covariant vector field.
This is called the
covariant
&,
and is denoted by
<ita.
More generally, a long but quite straightforward calculation analogous
to the above can be made to establish the
covariant derivative
a tensor
Till.::bp,p of rank
p
+
q,
contravariant of rank
p,
covariant of rank
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.
 Fall '05
 Thomson

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