326
DOC.
71
PRINCETON LECTURES
THE GENERAL THEORY
each
index;
as
with
vectors,
the
character
is designated
by
the
position
of
the
index.
For
example,
Auv
denotes
a
tensor
of the
second
rank,
which
is
co-variant with
respect
to
the index
u,
and
contra-variant with
respect to
the
index
v.
The
tensor
character indicates that
the
equation
of
transformation
is
(58)
a:'
=
dxa
dx'u
M-A'A°-
dxß
Tensors
may
be
formed
by
the
addition
and
subtraction
of
tensors
of
equal
rank
and like character,
as
in
the
theory
of
invariants
of
orthogonal
linear
substitutions,
for
example,
(59)
a;
+
b:
=
c;.
The
proof
of the
tensor
character
of
Cuv
depends
upon (58).
Tensors
may
be formed
by
multiplication,
keeping
the
character
of
the
indices,
just
as
in the
theory
of
invariants
of
linear
orthogonal transformations,
for
example,
(60)
a;b,t
=
c^.
The
proof
follows
directly
from
the rule
of
transformation.
Tensors
may
be formed
by
contraction
with
respect to
two
indices of
different
character,
for
example,
(61) A^r
=
B".
The
tensor
character
of
AuuoT
determines
the
tensor
character
of
BoT.
Proof-
,
_
fa*
dx\
dx^
ß
dx,
dx,
*“r
dx\
dxß
dx',
dx'T
Aa!t
dx',
dx'T
Aa,‘‘
The
properties
of
symmetry
and
skew-symmetry
of
a
tensor
with
respect to two
indices of
like character
have
the
same
significance as
in the
theory
of
special
relativity.
With
this,
everything
essential has
been
said
with
regard
to
the
algebraic properties
of
tensors.
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