Flat form
A measurable -dimensional differential form
on an open set
such that: 1) the co-mass (cf. Mass and co-mass)
for a given
; and 2) there exists an
with
![]() |
for any simplex satisfying the following condition: There exists a measurable
,
, such that
is measurable on
and on any one of its boundaries
, making up
; moreover,
![]() |
Here, denotes the
-dimensional Lebesgue measure of the intersection of the set
with some
-dimensional plane.
If is an
-dimensional flat cochain in
, there exists a bounded
-dimensional form
in
which is measurable in any simplex
with respect to the plane which contains
, and
![]() | (1) |
Also
![]() |
where is the co-mass of the cochain
. Conversely, to any
-dimensional flat form
in
there corresponds, according to formula (1), a unique
-dimensional flat cochain
for any simplex
which satisfies the above condition; moreover,
![]() |
The form and the cochain
are called associated. Forms associated with the same cochain are equivalent, i.e. are equal almost-everywhere in
, and comprise the flat representative.
There is a one-to-one correspondence between the -dimensional flat cochains
and the classes of equivalent bounded measurable functions
, given by
, and
![]() |
where is a sequence of
-dimensional simplices contracting towards the point
such that their diameters tend to zero, but such that
![]() |
for some value of , where
is the volume
for all
.
Let be a measurable summable function in
whose values are
-vectors; it is said to correspond to an
-dimensional flat chain if
![]() | (2) |
for all -dimensional flat cochains
(
is then called a Lebesgue chain). The mapping
is a linear one-to-one mapping of the set of equivalence classes of functions
into the space of flat chains
; also,
, where
is the mass of the chain
, (cf. Mass and co-mass) and
is the mass of the
-vector
. In addition, the set of images of continuous functions
is dense in
.
Formulas (1) and (2) generalize similar results for sharp forms and sharp cochains (cf. Sharp form); for instance, the differential of the flat form , defined by the formula
, is also a flat form, and Stokes' theorem:
is valid for any simplex
; an
-dimensional flat cochain is the weak limit of smooth cochains, i.e. cochains for which the associated forms
are smooth, etc.
References
[1] | H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) |
Flat form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flat_form&oldid=18201