1. Introduction
In [2], we have the Converse of the Poincar’e Lemma:
Lemma 1.1. Let U be a domain in 
which can be deformed to a point P. Let ω be a (p+1)-form on U such that
. Then there is a p-form 
in U such that
(1)
And in [1] we have Lemma 1.2. Let D be a bounded, convex domain in
. To each 
there corresponds a linear operator 
defined by
(2)
and the decomposition
(3)
holds at any point y in D.
In this paper, we extend the results of both of them. First we extend the bounded, convex domain D to the domain that deformed to every interior point. Then we not only gain that the closed form is the exact form, but every form can be decomposited to two parts where one of them is an exact form and another is a form related to the exterior differential of the form.
2. Preliminaries
It’s well-known that differential forms are the generalizations of the functions and have been applied to many fields such as potential theory, partial differential equations, quasiconformal mappings, nonlinear analysis, electromagnetism and control theory. First, we introduce some notations and preliminaries about differential forms. Let U denote an open subset of 
and
. Let 
denote the standard orthogonal basis of Rn. 
is the linear space of l-covectors, generated by the exterior products
, corresponding to all ordered l-tuples 
. The Grassman algebra 
is a graded algebra with respect to the exterior products.
A differential l-form ω on U is a Schwartz distribution on U with values in
. Let 
denote the space of all differential l-forms and the class of infinitely differentiable l-forms on U by
.
Then we define the mapping f* for a smooth mapping f on U into V, where U is a domain in 
and V is a domain in
, that is
(4)
We denote by 
the coordinates of Rm and by 
the coordinates of Rn. Then we can write
(5)
to show that the point with coordinates x is transformed by f to the point with coordinated y. The function
are smooth. Now we define the map f* taking l-forms on V to l-forms on U:
(6)
And there are basic properties for the mapping 
we’ll use in the following statement.
Lemma 2.1. If ω is a l-form on V, then
(7)
Lemma 2.2. If 
and
, then
(8)
More essential properties for 
can be found in [3]. More preliminaries of differential forms and their applications can be found in [1-15].
Then we define another important mapping:
Definition 2.1. Given a function 
is a continuous for (x, u) [3]. We call a domain U is deformable to a point p if there exists 
such that
(9)
(10)
Then we can analogously define that a domain is deformable to any point
, and denote the function 
as 
for every y.
3. Main Results and Proofs
First, we introduce the “cylinder construction”. Let U be a domain in 
that is deformable to any 
just like we have defined. We denote by [0, 1] the unit interval on the t-axis and consider the cylinder or product space 
This consists of all pairs (t, x) where 
and x runs over points of U. We point out the two maps which identify U with the top and bottom of the cylinder, that is
(11)
Thus
(12)
For example, to form 
where ω is a form on
, we simply replace t by 1 wherever it occurs in ω (and dt by 0 correspondingly). Now we form a new operation 
for any 
(13)
is defined on monomials by the formulas:
(14)
(15)
and on general differential forms by summing the results on the monomial parts. Here is the basic property of
:
Lemma 3.1. If ω is any (p+1)-form on
, then
(16)
Proof: We only need to check this for monomials.
Case 1. 
We have 
(17)
(18)
But 
So the formula is valid .
Case 2. 
First notice 
Next we have
, (19)
(20)
So the formula works, again.
We can easily get the following conclusion:
Lemma 3.1. For the mapping
, the boundary conditions may be interpreted in terms of the
as follows:
(21)
if U is deformable to the point y.
For an (l +1)-form ω on U we have
(22)
Now we state and prove the main result.
Theorem 3.1. Assume U is a domain in 
which can be deformed to every point
. Let ω be an (l + 1)-form on U. Then there is
(23)
Proof: We only substitute 
in the above formula of Lemma 3.1. And with Equation (22), we finish the proof.
Thus we finish the extension. It’s interesting to see if
, then
. Hence with the formula above we have 
where 
This is just the generalization of the converse of the Poincar’e Lemma in [3], which shows that closed form is an exact form.
Corollary 3.1. Assume U is a domain in 
which can be deformed to every point
. If ω is a closed (l+1)-form on U, then it is an exact form. Then we can construct a homotopy operator 
by averaging 
over all points
:
(24)
where 
in 
is normalized so that
It is obvious that the main result of this article remains valid for the operator T:
(25)
We begin with the equation of Lemma 3.1
(26)
Multiplying 
to it and integraling on U, we have
(27)
Then with 
we obtain
(28)
which yields the above formula.
4. Conclusion
We have obtained an extension of the Poincar’e Lemma for differential forms in a bounded, convex domain in Rn to a more general domain. Then we have extended the homotopy operator T to the domain defromed to every point of itself. So all of the conclusions about the homotopy operator T can be extended to the deformed domain.
5. Acknowledgements
The research of the author was supported by the Fundamental Research (2010) of NUDT (NO. JC10-02-02).