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
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
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:
where is a polyhedral chain and is the value of the cochain on the chain .
For references see Flat norm.
|[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://www.encyclopediaofmath.org/index.php?title=Mass_and_co-mass&oldid=28243