Mass and co-mass
Adjoint norms (cf. Norm) in certain vector spaces dual to each other.
1) The mass of an -vector
, i.e. an element of the
-fold exterior product of a vector space, is the number
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The co-mass of an -covector
is the number
![]() |
Here is the standard norm of an
-vector and
is the scalar product of a vector and a covector.
The mass and the co-mass
are adjoint norms in the spaces of
-vectors
and
-covectors
, respectively. In this connection:
a) ,
;
b) ,
, and equalities hold if and only if
(
) is a simple
-(co)vector;
c) ,
for exterior products
, where for a simple multi-covector
(or
)
, and, in general,
if
and
;
d) for inner products
, where
for
and
for
,
and
.
These definitions enable one to define the mass and co-mass for sections of fibre bundles whose standard fibres are and
. For example, the co-mass of a form
on a domain
is
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2) The mass of a polyhedral chain is
![]() |
where is the volume of the cell
. For arbitrary chains the mass (finite or infinite) can be defined in various ways; for flat chains (see Flat norm) and sharp chains (see Sharp norm) these give the same value to the mass.
3) The co-mass of a (flat, in particular, sharp) cochain is defined in the standard way:
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where is a polyhedral chain and
is the value of the cochain
on the chain
.
For references see Flat norm.
Comments
A simple -vector
is an element of the form
in the
-fold exterior product
of a vector space
. Here "" denotes exterior product and
.
References
[a1] | H. Federer, "Geometric measure theory" , Springer (1969) pp. Sect. 1.8 MR0257325 Zbl 0176.00801 |
Mass and co-mass. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mass_and_co-mass&oldid=47781