§ 3. Integration of Differential Forms
189
(x; h 1 , . . . , h p ) =
ω (tx; x, h 1 , . . . , h p ) t
p dt .
(8.14)
These calculations show that locally, any differential closed form of degree p
is exact, i.e. is the exterior derivative of a form of degree p − 1 ; the latter
is not unique: a closed form, i.e. as we are using local arguments, an exact
differential can always be added to it.
This argument still holds globally if G is, for example, star shaped, and
hence convex. But the general case is far more difficult to deal with, even
in a simply connected domain if forms of degree ≥ 2 are considered ; the
problem is directly connected to the topology of G (De Rham cohomology);
the following simple argument may give a vague idea.
G can always be written as the finite or infinite union of open convex
non-empty subsets U i ; the intersections
U ij = U i ∩ U j , U ijk = U i ∩ U j ∩ U k , etc. ,
remain convex (and possibly empty). Then take for example a closed form ω
of degree 3 on G, de degr´ e 3. There are forms ω i of degree 2 on U i such that
ω = dω i on U i .
As dω i = dω j on U ij , there are forms ω ij of degree 1 on the non-empty U ij
such that
ω i − ω j = dω ij on U ij .
Then d(ω jk − ω ik + ω ij ) = 0 on each U ijk , and so there are forms ω ijk of
degree 0 (i.e functions) on the non-empty U ijk such that
ω jk − ω ik + ω ij = dω ijk on U ijk .
So
d(ω jkh − ω ikh + ω ijh − ω ijk ) = 0 on U ijkh ,
and as the non-empty U ijkh are convex, the relations
ω jkh − ω ikh + ω ijh − ω ijk = c ijkh
follow, where the c ijkh are constants associated to the non-empty U ijkh and
satisfying
c jkhl − c ikhl + c ijhl − c ijkl + c ijkh = 0
whenever U ijkhl is non-empty. Thus the scheme of non-empty pairwise, threewise, etc. intersections of the open subsets U i – he “ simplicial structure ”
of the cover considered – is involved, and this takes us into the realm of
189
(x; h 1 , . . . , h p ) =
ω (tx; x, h 1 , . . . , h p ) t
p dt .
(8.14)
These calculations show that locally, any differential closed form of degree p
is exact, i.e. is the exterior derivative of a form of degree p − 1 ; the latter
is not unique: a closed form, i.e. as we are using local arguments, an exact
differential can always be added to it.
This argument still holds globally if G is, for example, star shaped, and
hence convex. But the general case is far more difficult to deal with, even
in a simply connected domain if forms of degree ≥ 2 are considered ; the
problem is directly connected to the topology of G (De Rham cohomology);
the following simple argument may give a vague idea.
G can always be written as the finite or infinite union of open convex
non-empty subsets U i ; the intersections
U ij = U i ∩ U j , U ijk = U i ∩ U j ∩ U k , etc. ,
remain convex (and possibly empty). Then take for example a closed form ω
of degree 3 on G, de degr´ e 3. There are forms ω i of degree 2 on U i such that
ω = dω i on U i .
As dω i = dω j on U ij , there are forms ω ij of degree 1 on the non-empty U ij
such that
ω i − ω j = dω ij on U ij .
Then d(ω jk − ω ik + ω ij ) = 0 on each U ijk , and so there are forms ω ijk of
degree 0 (i.e functions) on the non-empty U ijk such that
ω jk − ω ik + ω ij = dω ijk on U ijk .
So
d(ω jkh − ω ikh + ω ijh − ω ijk ) = 0 on U ijkh ,
and as the non-empty U ijkh are convex, the relations
ω jkh − ω ikh + ω ijh − ω ijk = c ijkh
follow, where the c ijkh are constants associated to the non-empty U ijkh and
satisfying
c jkhl − c ikhl + c ijhl − c ijkl + c ijkh = 0
whenever U ijkhl is non-empty. Thus the scheme of non-empty pairwise, threewise, etc. intersections of the open subsets U i – he “ simplicial structure ”
of the cover considered – is involved, and this takes us into the realm of