280 DOC.
71
PRINCETON LECTURES
PRE-RELATIVITY PHYSICS
also be
a
vector
it
is
necessary
for
pvo
to
be
a
tensor.
Then by
differentiation and contraction
dpvo/dxo
results,
and
is
therefore
a
vector,
as
it also
is
after
multiplication
by
the
reciprocal
scalar
1/p.
That
pvo
is
a
tensor,
and therefore
transforms
according
to
the
equation
Pß»=
bnabyßpaßl
is
proved
in
mechanics
by
integrating
this
equation
over
an
infinitely
small
tetrahedron. It
is
also
proved
there,
by application
of
the theorem
of
moments to
an
infinitely
small
parallelepipedon,
that
pvo =
pov,
and hence that
the
tensor
of the
stress is
a
symmetrical tensor.
From what
has
been
said it follows
that,
with the
aid of
the
rules
given
above,
the
equation
is
co-variant with
respect to
orthogonal
transformations in
space (rotational trans-
formations);
and
the
rules
according
to
which the
quanti-
ties in
the
equation
must
be
transformed
in
order that the
equation may
be
co-variant
also
become evident.
The co-variance
of
the
equation
of
continuity,
(17)
dp d(pu,)
_ dt
+
dx.
requires,
from
the
foregoing,
no
particular
discussion.
We shall also
test
for
co-variance
the
equations
which
express
the
dependence
of
the
stress
components upon
the
properties
of the
matter,
and
set up
these
equations
for
the
case
of
a
compressible
viscous fluid
with the aid
of
the conditions
of
co-variance.
If
we
neglect
the
vis-
cosity,
the
pressure, p,
will be
a
scalar,
and
will
depend
only upon
the
density
and the
temperature
of
the
fluid.
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