DOC.
71
PRINCETON LECTURES 325
THE GENERAL THEORY
the
place
for
a
brief
presentation
of
the
most important
mathematical
concepts
and
operations
of this
calculus of
tensors.
We
designate
four
quantities,
which
are
defined
as
functions of the
xv
with
respect to
every system
of coordi-
nates,
as
components,
Av,
of
a
contra-variant
vector, if
they
transform
in
a
change
of co-ordinates
as
the co-ordinate
differentials
dxv.
We
therefore have
(56) Au'
=
A-
Besides
these
contra-variant
vectors,
there
are
also
co-
variant
vectors.
If
Bv are
the
components
of
a
co-variant
vector,
these
vectors
are
transformed
according
to
the rule
(57)
B'u
-
Bv.
The definition of
a
co-variant
vector
is
chosen in
such
a
way
that
a
co-variant
vector
and
a
contra-variant
vector
together
form
a
scalar
according to
the
scheme,
f
=
BvAv
(summed
over
the
v).
For
we
have
ß'-A"
-
B.A-
=
B.A-.
In
particular,
the derivatives
dQ/dxa
of
a
scalar
0,
are com-
ponents
of
a
co-variant
vector, which,
with the co-ordinate
differentials,
form
the scalar
dQ/dxa
dxa; we see
from this
example
how natural
is
the definition
of
the co-variant
vectors.
There
are
here,
also,
tensors
of
any
rank,
which
may
have co-variant
or
contra-variant character with
respect to
[65]