From (8) we obtain after multiplication by
hence
is still defined as
, as in (19
). From (55
) it follows that the concavity of
- the negative definiteness of
- implies global invertibility between the field vector
and the Lagrange multipliers
,
provided that
the privileged co-vector
is chosen as co-linear to the vector potential
. We set
Indeed, in that case we have
hence
so that, by (57), the second term on the right hand side of (55
) vanishes and
is definite. Equation (58
) will be used later.
With
as a field vector, instead of
, we may rephrase (8
) in the form
or
hence
where
. Thus
is the Legendre transform of
with respect to the map
. It follows that
is concave in
, since
is concave in
; thus we have
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Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |