A Mathematical Journal
Vol. 7, No 2, (139 - 169). August 2005.

Differential Forms and/or Multi-vector
Functions

F. Brackx
Department of Mathematical Analysis

Ghent University. Galglaan 2, B-9000 Gent, Belgium
fb@cage.ugent.be

R. Delanghe
Department of Mathematical Analysis

Ghent University. Galglaan 2, B-9000 Gent, Belgium
rd@cage.ugent.be

F. Sommen
Department of Mathematical Analysis

Ghent University. Galglaan 2, B-9000 Gent, Belgium
fs@cage.ugent.be

”... on peut se poser la question: quel est le théorème
mathématique le plus profond, le plus difficile, dont il existe
une interprétation physique concrète et indubitable? (...) Pour
moi, c’est le théorème de Stokes qui est le candidat numéro un.
Et cela témoigne d’un fait: la différentielle extérieure est une
notion très mystérieuse, dont la véritable nature, je crois, recèle
encore bien des énigmes, et cela en dépit de la simplicité de sa
définition formelle.”

René Thom, La science malgré tout ...

ABSTRACT
Similarities are shown between the algebras of differential forms and of Clif-

ford algebra-valued multi-vector functions in an open region of Euclidean space.
The Poincaré Lemma and the Dual Poincaré Lemma are restated and proved in a
refined version. In the case of real-analytic differential forms an alternative proof
of the Poincaré Lemma is given using the Euler operator. A position is taken in



140 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

the debate on the redundancy of either of the two algebras.

RESUMEN

Se muestran similitudes entre las álgebras de formas diferenciales y las de fun-
ciones multivectoriales valuadas de una álgebra de Clifford en una región abierta
del espacio Euclidiano. El Lema de Poincaré y Lema de Poincaré dual son pre-
sentados y probados en una versión refinada. En el caso de formas diferenciales
reales anaĺıticas una prueba alternativa del Lema de Poincaré es dada usando
el operador de Euler. Una posición es tomada en el debate en redundancia de
cualquiera de las dos álgebras.

Key words and phrases: differential forms, multi-vector functions,
Poincaré Lemma

Math. Subj. Class.: 58A10, 30G35

1 Introduction

In this paper two mathematical languages are confronted with each other: the lan-
guage of differential forms and the one of Clifford algebra-valued multi-vector func-
tions.
The Cartan algebra

∧
(Ω) of smooth differential forms on an open subset Ω of Euclidean

space Rm+1, endowed with exterior multiplication, is of course well-known. A fun-
damental operator on

∧
(Ω) is the exterior derivative d with its important property

that for any differential form ω, d(dω) = 0.

Introducing the Hodge co-derivative d∗ leads to the differential operator D =
d + d∗, by means of which the so-called ”harmonic” r-forms (0 < r < m + 1) are
characterized as smooth differential r-forms ωr satisfying Dωr = 0.
The algebra E(Ω) of smooth multi-vector functions is less well-known. Multi-vector
functions arise in a natural way when considering functions defined in Ω and tak-
ing values in the universal real Clifford algebra R0,m+1 constructed over R0,m+1, i.e.
Rm+1 equipped with an anti-Euclidean metric. If Rr0,m+1 (0 ≤ r ≤ m + 1) denotes
the space of r-vectors, then the Clifford algebra R0,m+1 is precisely the graded as-
sociative algebra R0,m+1 =

∑m+1
r=0 ⊕ Rr0,m+1, and an r-vector function Fr is a map

Fr : Ω → Rr0,m+1. It was William Kingdon Clifford who introduced his so-called
geometric algebra in the 1870s, building on earlier work of Hamilton and Grassmann.
A fundamental operator on the space of smooth multi-vector functions, is the Dirac
operator ∂, by means of which the so-called monogenic functions are characterized
as the smooth functions f satisfying ∂f = 0. Note that the monogenic functions are



7, 2(2005)
Differential Forms and/or Multi-vector Functions 141

at the core of so-called Clifford analysis, a function theory which developed exten-
sively during the last decades, offering a direct and elegant generalization to higher
dimension of the theory of holomorphic functions in the complex plane. Note also
that the above mentioned Dirac equation may be expressed in the language of sys-
tems of partial differential equations by modelling Clifford algebra through its matrix
representation.
The spaces of smooth differential forms on the one hand, and of smooth multi-vector
functions on the other, are shown to be isomorphic in a natural way: a smooth r-form
is identified with a smooth r-vector function, the action of the differential operator
D = d + d∗ on the space

∧r(Ω) of smooth r-forms, is identified with the action of the
Dirac operator ∂ on the space Er(Ω) of smooth r-vector functions, and the counter-
parts in the space of multi-vectors of the exterior derivative d and the co-derivative
d∗ are pinpointed. This isomorphism is moreover fully exploited in that proofs can
be given in either of both languages and that the results obtained are mutually ex-
changeable (section 4).
In fact the paper also focusses on two well-known theorems on differential forms: the
Poincaré Lemma and the Dual Poincaré Lemma. They are restated in a refined ver-
sion which, to the authors’ knowledge, rarely appears in the literature. Combining
these two theorems, a structure theorem for monogenic multi-vector functions and its
counterpart in the space of smooth differential forms is given (section 5). In proving
these structure theorems, we heavily rely on the classical Poincaré Lemma and the
classical Dual Poincaré Lemma. In section 6 an alternative proof of those lemmata
are given in the special case of real-analytic differential forms in an open ball centred
at the origin.
We wish to emphasize that the present paper may not be seen as a pleading to sub-
stitute one of the languages for the other, nor to prefer one language above the other.
On the contrary, we are convinced that differential forms and multi-vector functions,
despite the natural identification given, are quite different mathematical objects, the
use of which is very much imposed by the mathematical context. This in-depth differ-
ence between and context-dependence of differential forms and multi-vector functions
will be fully discussed in a forthcoming paper by one of the authors.

2 Multi-vector functions: preliminaries

In this section we recall some basic notions and results from Clifford algebra and
Clifford analysis. For a detailed account we refer the reader to [10] and [2]; the re-
cent book [3] gives a nice and broad overview of the intrinsic value and usefulness of
Clifford algebra and Clifford analysis for mathematical physics.
The construction of the universal real Clifford algebra is well-known; we restrict our-
selves to a schematic approach. Let R0,m+1 be the real vector space Rm+1 (m ≥ 1)
endowed with a non-degenerate symmetric bilinear form B of signature (0,m + 1),
and let (e0,e1, · · · ,em) be an associated orthonormal basis:

B(ei,ej ) =
{
−1 if i = j

0 if i �= j (0 ≤ i,j ≤ m).



142 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

The anti-Euclidean metric on R0,m+1 is induced by the scalar product

< ei,ej >= −B(ei,ej) = δij, 0 ≤ i,j ≤ m.
Introduce the anti-symmetric outer product by the rules:

ei ∧ ei = 0, 0 ≤ i ≤ m
ei ∧ ej + ej ∧ ei = 0, 0 ≤ i �= j ≤ m.

For each A = {i1, i2, · · · , ir} ⊂ M = {0, 1, · · · ,m}, ordered in the natural way:
0 ≤ i1 < i2 < · · · < ir ≤ m, put

eA = ei1 ∧ ei2 ∧ ·· · ∧ eir
and

eφ = 1.

Then for each r = 0, 1, · · · ,m + 1, the set {eA : A ⊂ M and |A| = r} is a basis for
the space Rr0,m+1 of so-called r-vectors.

Introducing the inner product by

ei • ej = − < ei,ej >, 0 ≤ i,j ≤ m
leads to the so-called geometric product in the Clifford algebra, given by

eiej = ei • ej + ei ∧ ej, 0 ≤ i,j ≤ m.
The respective definitions of the inner product, the outer product and the (geometric)
product are then extended to r-vectors by the formulae:

ej • eA = ej • (ei1 ∧ ·· · ∧ eir ) =
∑

k

(−1)kδjik eA\{ik}

where
eA\{ik} = ei1 ∧ ·· · ∧ eik−1 ∧ [eik ∧] eik+1 ∧ ·· · ∧ eir

and{
ej ∧ eA = ej ∧ (ei1 ∧ ·· · ∧ eir ) = ej ∧ ei1 ∧ ·· · ∧ eir, if j /∈ A
ej ∧ eA = 0, if j ∈ A

and finally
ejeA = ej • eA + ej ∧ eA.

The inner and outer products are distributive over addition, and so is the (geometric)
product.
The universal real Clifford algebra R0,m+1 is the graded associative algebra

R0,m+1 =
m+1∑
r=0

⊕ Rr0,m+1.



7, 2(2005)
Differential Forms and/or Multi-vector Functions 143

If [ . ]r : R0,m+1 → Rr0,m+1 denotes the projection operator from R0,m+1 onto Rr0,m+1,
then each Clifford number a ∈ R0,m+1 may be written as

a =
m+1∑
r=0

[a]r.

Note that in particular for a 1-vector u and an r-vector vr , one has

u vr = u • vr + u ∧ vr
with

u • vr = [u vr]r−1 =
1
2

(
u vr − (−1)rvr u

)

and

u ∧ vr = [u vr]r+1 =
1
2

(
u vr + (−1)rvr u

)
.

Usually R and Rm+1 are identified with R00,m+1 and R
1
0,m+1 respectively. An element

x = (x0,x1, · · · ,xm) ∈ Rm+1 is thus identified with the 1-vector x =
∑m

j=0 xj ej .

Now let Ω be an open region in Rm+1. A smooth r-vector function Fr is a map

Fr : Ω → Rr0,m+1, x 
→
∑
|A|=r

Fr,A(x) eA

where for each A, Fr,A is a smooth real-valued function in Ω.
We denote by Er(Ω) the space of smooth r-vector functions in Ω, and we put

E(Ω) =
m+1∑
r=0

⊕ Er(Ω).

The projection operator from E(Ω) onto Er(Ω) is denoted by [ . ]r. For the linear
operator T : Er(Ω) →E(Ω) we denote by

r

ker T the kernel of T in Er(Ω), while
r

im T
stands for the image of Er(Ω) under T .

A fundamental operator in Clifford analysis is the so-called Dirac operator, a vector
differential operator given by

∂ =
m∑

j=0

ej ∂xj .

Due to the non-commutativity of the multiplication in the Clifford algebra, it can act
from the left or from the right on a function. For F =

∑
A

eAFA ∈E(Ω) these actions
are given by

∂F =
∑

j

∑
A

ejeA ∂xj FA and F∂ =
∑

J

∑
A

eAej ∂xj FA.



144 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

A function F ∈E(Ω) is called left (resp. right) monogenic in Ω iff it satisfies ∂F = 0
(resp. F∂ = 0) in Ω.

Restricting the Dirac operator ∂ to the space Er(Ω) we find for an r-vector function
Fr, that ∂Fr and Fr∂ split up into an (r− 1)-vector and an (r + 1)-vector function:

∂Fr =
m∑

j=0

ej ∂xj Fr =
∑

j

ej • ∂xj Fr +
∑

j

ej ∧ ∂xj Fr
and

Fr∂ =
m∑

j=0

∂xj Fr ej =
∑

j

∂xj Fr • ej +
∑

j

∂xj Fr ∧ ej.

It readily follows that

[∂Fr]r−1 =
∑

j

ej • ∂xj Fr = (−1)r+1
∑

j

∂xj Fr • ej = (−1)r+1[Fr∂]r−1

[∂Fr]r+1 =
∑

j

ej ∧ ∂xj Fr = (−1)r
∑

j

∂xj Fr ∧ ej = (−1)r[Fr∂]r+1.

Consequently, for an r-vector function Fr , the notions of left monogenicity and right
monogenicity coincide.
Moreover, if for F ∈E(Ω) we put FE =

∑
|A|=even

eA FA and FO =
∑

|A|=odd
eA FA, then

F is monogenic in Ω iff both FE and FO are monogenic in Ω.

Commonly one introduces the notations:

∂ • Fr = [∂Fr]r−1 , ∂ ∧ Fr = [∂Fr]r+1
Fr • ∂ = [Fr∂]r−1 , Fr ∧ ∂ = [Fr∂]r+1.

The action of the Dirac operator ∂ on Er(Ω) thus gives rise to two auxiliary differential
operators:

∂− : Er(Ω) → Er−1(Ω) : Fr 
→ ∂−Fr = ∂ • Fr = [∂Fr]r−1
and
∂+ : Er(Ω) → Er+1(Ω) : Fr 
→ ∂+Fr = ∂ ∧ Fr = [∂Fr]r+1.

Symbolically these operators may be written as:

∂− = (∂ • ) =
∑

j

(ej • )∂xj

and
∂+ = (∂ ∧) =

∑
j

(ej ∧)∂xj .



7, 2(2005)
Differential Forms and/or Multi-vector Functions 145

Their action on Er(Ω) is two-fold in the sense that they act on the multi-vector by
means of the inner and outer product with basis vectors, and at the same time on the
function coefficients by partial differentiation.

As on Er(Ω) holds:
∂ = ∂− + ∂+

we obtain that a smooth r-vector function Fr is left monogenic (as well as right
monogenic) in Ω iff in Ω

∂Fr = 0 ⇐⇒ Fr∂ = 0 ⇐⇒
{
∂−Fr = 0
∂+Fr = 0

. (I)

As the Dirac operator ∂ splits the Laplace operator:

∂2 = ∂ • ∂ + ∂ ∧ ∂ = ∂ • ∂ = − < ∂,∂ > = −

a monogenic function in Ω is also harmonic in Ω, but the converse clearly is not true.

As moreover
(∂−)2 = (∂+)2 = 0

we have
−
 = (∂− + ∂+)2 = ∂−∂+ + ∂+∂−.

The second order differential operators ∂−∂+ and ∂+∂− are scalar operators in the
sense that they keep the order of the multi-vector function, but the function coeffi-
cients, while being differentiated, are interchanged w.r.t. the basis multi-vectors.

Now observe that the system (I), expressing the monogenicity of an r-vector func-
tion, is also equivalent to

∂̃Fr = (∂+ −∂−)Fr = 0
or

Fr∂̃ = Fr (∂+ −∂−) = 0
where we have introduced the modified Dirac operator

∂̃ = ∂+ −∂−.

We directly have the basic formulae:

∂∂̃ = ∂−∂+ −∂+∂−

∂̃∂ = ∂+∂− −∂−∂+

∂̃∂̃ = −∂+∂− −∂−∂+ = 



146 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

which leads to the modified Laplace operator


̃ = ∂−∂+ −∂+∂−

which clearly is a scalar operator in the sense that it keeps the order of the multi-
vector function on which it acts.

Taking into account the main involution, also called inversion, of the Clifford
algebra, for which

(ei1 · · · eir )∗ = (ei1 ∧ ·· · ∧ eir )∗ = (−1)r ei1 ∧ ·· · ∧ eir
we get the formulae:

∂Fr = F∗r ∂̃ and ∂̃Fr = F
∗
r ∂

−
Fr = ∂∂Fr = ∂F∗r ∂̃ = (−1)r ∂Fr∂̃

Fr = ∂̃∂̃Fr = ∂̃F∗r ∂ = (−1)r ∂̃Fr∂

̃Fr = ∂∂̃Fr = ∂F∗r ∂ = (−1)r ∂Fr∂
−
̃Fr = ∂̃∂Fr = ∂̃F∗r ∂̃ = (−1)r ∂̃Fr∂̃.

3 Differential forms: preliminaries

This section is also introductory; there is a vast literature on differential forms; we
may refer to e.g. [8], [15].
Let Rm+1 be endowed with the standard Euclidean metric.
Denoting by

∧r
Rm+1 the space of alternating (or skew-multilinear) real-valued r-

forms (0 ≤ r ≤ m + 1), the Grassmann algebra or exterior algebra over Rm+1 is the
graded associative algebra

∧
Rm+1 =

m+1∑
r=0

⊕
∧r

Rm+1

endowed with the exterior multiplication.
A basis for

∧r
Rm+1 is obtained as follows. Let {dx0,dx1, · · · ,dxm} be a basis for the

dual space (Rm+1)∗ of Rm+1. If again the set A = {i1, . . . , ir} ⊂ M = {0, 1, · · · ,m}
is ordered in the natural way, put

dxA = dxi1 ∧ dxi2 ∧ ·· · ∧ dxir

and
dxφ = 1.



7, 2(2005)
Differential Forms and/or Multi-vector Functions 147

Then for each r = 0, 1, · · · ,m + 1, the set {dxA : A ⊂ M and |A| = r} is a basis for∧r
Rm+1.

Note that in particular

dxi ∧ dxi = 0, i = 0, 1, · · · ,m + 1
and

dxi ∧ dxj + dxj ∧ dxi = 0, 0 ≤ i �= j ≤ m.
A smooth r-form in an open region Ω of Rm+1 is a map

ωr : Ω →
∧r

Rm+1, x 
→
∑
|A|=r

ωrA(x) dx
A

where for each A, ωrA is a smooth real-valued function in Ω.

We denote by
∧r(Ω) the space of smooth r-forms in Ω and we put

∧
(Ω) =

m+1∑
r=0

⊕
∧r

(Ω).

The projection operator from
∧

(Ω) onto
∧r(Ω) is denoted by [ . ]r, and the notations

of the foregoing section are kept for the kernel and the image of a linear operator
T :
∧r(Ω) −→ ∧(Ω).
A fundamental linear operator on the space of smooth forms is the exterior deriv-

ative d. It is first defined on
∧r(Ω) (r < m + 1) by
d :

∧r
(Ω) −→

∧r+1
(Ω)

ωr =
∑
|A|=r

ωrA dx
A 
−→ dωr =

∑
A

∑
j

∂xj ω
r
A dx

j ∧ dxA

and this definition is then extended to
∧

(Ω) by linearity.
The kernel of the exterior derivative d ,

r

ker d = {ωr ∈
∧r

(Ω) : dωr = 0}

consists of the so-called closed r-forms in Ω, while its image of
∧r−1(Ω) in ∧r(Ω)

r−1
im d = {dωr−1 : ωr−1 ∈

∧r−1
(Ω)}

consists of the so-called exact r-forms in Ω.

The quotient space

Hr(Ω) =
r

ker d /
r−1
im d



148 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

is the so-called de Rham r-th cohomology space.

The well-known Poincaré Lemma (see also section 5) asserts that if Ω is con-
tractible to a point, then for each r > 0, Hr(Ω) = 0, in other words: if Ω is con-
tractible to a point and ωr ∈ ∧r(Ω) is closed, then ωr is exact. The converse, i.e.
that any exact r-form in an open region of Rm+1 is also closed, follows at once from
the observation that d(dω) = 0.

A second fundamental linear operator on the space of smooth forms is the Hodge
co-derivative d∗. For A = {ii, · · · , ir}⊂ M we denote

dxA\{ij } = dxi1 ∧ ·· · ∧ dxij−1 ∧ [dxij ∧] dxij+1 ∧ ·· · ∧ dxir

and in a first step we put:

d∗(ωAdx
A) =

r∑
j=1

(−1)j ∂xj ωA dxA\{ij }.

Then d∗ is defined on
∧r(Ω) (r > 0) by

d∗ :
∧r

(Ω) −→
∧r−1

(Ω)

ωr =
∑
|A|=r

ωrA dx
A 
−→ d∗(ωr) =

∑
|A|=r

d∗(ωrA dx
A)

and this definition is extended to
∧

(Ω) by linearity.
The kernel of the co-derivative d∗ acting on

∧
(Ω):

r

ker d∗ = {ωr ∈
∧r

(Ω) : d∗ωr = 0}

consists of the so-called co-closed r-forms in Ω, while its image of
∧r+1(Ω) in ∧r(Ω)

r+1
im d∗ = {d∗ωr+1 : ωr+1 ∈

∧r+1
(Ω)}

consists of the so-called co-exact r-forms in Ω.
By observing that for any smooth form in Ω, d∗(d∗ω) = 0, it follows that each co-exact
r-form in Ω is also co-closed. The quotient space

Hr(Ω) =
r

ker d∗ /
r+1
im d∗

is the so-called de Rham r-th homology space.
It could be confusing to use the term ”homology” here, since it usually refers to the
complex associated with the algebra of chains subject to the action of the boundary-
operator; in the space of currents however there is a connection (see [8], p.313).



7, 2(2005)
Differential Forms and/or Multi-vector Functions 149

By virtue of the Weyl duality we have for a region Ω which is contractible to a
point, and for each r < m + 1, that Hr(Ω) = 0, in other words: if Ω is contractible
to a point, then each co-closed r-form in Ω is also co-exact; this is dealt with in the
so-called Dual Poincaré Lemma (see section 5).

A smooth r-form in Ω which is at the same time closed and co-closed is called
harmonic in Ω (in the sense of Hodge). Introducing the operator D = d + d∗, a
necessary and sufficient condition for a smooth r-form ωr in Ω to be harmonic in Ω
thus reads:

Dωr = (d + d∗)ωr = 0 ⇐⇒
{
dωr = 0
d∗ωr = 0

. (II)

The system (II) is called the Hodge-de Rham system.

Note that if ωr is harmonic in an open region Ω of Rm+1 then automatically ωr

satisfies 
ωr = 0 in Ω, since
D2 = (d + d∗)2 = d d∗ + d∗ d = −
 .

The converse is however not true.

The action of the operators d and d∗ on differential forms is two-fold in the sense
that they act on the form itself as well as on the function coefficients by partial
differentiation. In order to explicit this double action we introduce the following
symbolic notations for the operators d and d∗. For d the following notation is rather
obvious:

d =
m∑

j=0

(dxj∧) ∂xj .

We then indeed have

dωr =

⎛
⎝∑

j

(dxj∧)∂xj

⎞
⎠
⎛
⎝ ∑

|A|=r
ωrA dx

A

⎞
⎠

=
∑

j

∑
A

∂xj ω
r
A dx

j ∧dxA

illustrating the above mentioned double action and the fact that d acts in an ”exterior”
way.
But this raises the question whether there exists a differential operator on forms acting
in an ”inner” way, to which end an ”inner product” in the Grassmann algebra should
be defined. Inspired by the inner product in the Clifford algebra, we put by definition:

dxi • dxj = − < dxi,dxj > = −δij, 0 ≤ i,j ≤ m.
In fact this scalar product in the Grassmann algebra already tacitly exists. Indeed,
as Rm+1 is endowed with the standard Euclidean metric, there is a canonical isomor-
phism between the tangent space TxΩ ∼= Rm+1 and its dual T ∗x Ω ∼= (Rm+1)∗, given



150 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

by
ej

iso←→ < ej, • > = e∗j = dxj

and hence

< dxi,dxj > = < e∗i ,e
∗
j > = < ei,ej > = δij, 0 ≤ i,j ≤ m.

So we introduce the operator
m∑

j=0

(dxj • ) ∂xj

clearly an operator with a double action.

In a next step we put

dxj • dxA = dxj • (dxi1 ∧ ·· · ∧ dxir ) =
r∑

k=1

(−1)k δjik dxA\{ik}.

We then get, by linearity, for a smooth r-form ωr:
⎛
⎝ m∑

j=0

(dxj • )∂xj

⎞
⎠
⎛
⎝ ∑

|A|=r
ωrAdx

A

⎞
⎠ =

r∑
k=1

∑
|A|=r

(−1)k (∂xik ω
r
A) dx

A\{ik }

in which we recognize the action of the co-derivative d∗ on ωr.
Consequently this co-derivative may be written as:

d∗ =
m∑

j=0

(dxj • )∂xj

also nicely illustrating the double action of d∗. From this point of view the co-
derivative d∗ might as well have been called ”interior derivative”.

Finally for the operator D = d + d∗ we obtain the expressions

D = d + d∗ =
m∑

j=0

(dxj∧ )∂xj +
m∑

j=0

(dxj • )∂xj

=
m∑

j=0

(dxj ∧ + dxj • )∂xj

=
m∑

j=0

(Dxj∨ )∂xj

where
Dxj∨ = dxj • + dxj∧



7, 2(2005)
Differential Forms and/or Multi-vector Functions 151

is the so-called ”vee-product”-operator, which was introduced in e.g. [7] and [12] in
the more general context of a metric with (p,q)-signature on Rm+1.

In the sequel we will deal with the operators d and d∗ on the same footing and
systematically mention the properties of d∗ next to those of d, for the sake of aes-
thetical symmetry. However, from the mathematical point of view this is superfluous;
considering the operator d∗ only leads to new results when it appears in connection
with the operator d. Note in this context the interesting operators dd∗ and d∗d, which
are the ”components” of the Laplace operator (−
).

4 Differential forms and multi-vector functions: an

identification

In becomes clear from sections 2 and 3 that the world of differential forms in an open
region Ω of Rm+1 and the world of multi-vector functions in Ω, may be identified in
a natural way. If for each A ⊂ M, fA is a smooth real-valued function in Ω, then the
following correspondence table may already be drawn (see next page).

This identification is now further developed. First one may wonder what the
counterpart is of the Hodge ∗ (star) operator. On the one hand one has

∗
(
dxj1 ∧ ·· · ∧ dxjr

)
= σ dxjr+1 ∧ ·· · ∧ dxjm+1

where j1 < · · · < jr , jr+1 < · · · < jm+1 , {j1, · · · ,jr}∪{jr+1, · · · ,jm+1} = M =
{0, 1, · · · ,m} and σ is the signature of the permutation (jr+1, · · · ,jm+1,j1, · · · ,jr).
This corresponds, for A = {j1, · · · ,jr}⊂ M to

∗eA = (−1)r eMe†A

where eM = e0 ∧ e1 ∧ ·· · ∧ em is the so-called pseudoscalar and † stands for the
main anti-involution of the Clifford algebra, also called reversion, given by

e
†
A = (ej1 ∧ ·· · ∧ ejr )

† = ejr ∧ ·· · ∧ ej1 = (−1)
r(r−1)

2 eA.

Next we identify some differential operators and establish similar formulae in both
worlds.

To start with, the Euler operator

E =
m∑

j=0

xj ∂xj



152 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

dxj ej

dxi ∧dxj ei ∧ej

dxi • dxj ei • ej

ωr =
∑
|A|=r

fA dx
A Fr =

∑
|A|=r

fA eA

d =
m∑

j=0

(dxj∧)∂xj ∂+ =
m∑

j=0

(ej∧)∂xj

d∗ =
m∑

j=0

(dxj •)∂xj ∂
− =

m∑
j=0

(ej •)∂xj

D = d + d∗ =
m∑

j=0

(Dxj∨)∂xj ∂ = ∂+ + ∂− =
m∑

j=0

ej ∂xj

ωr harmonic in Ω ⊂ Rm+1 Fr monogenic in Ω ⊂ Rm+1

d2 = dd = 0 ∂+2 = ∂+∂+ = 0

d∗2 = d∗d∗ = 0 ∂−2 = ∂−∂− = 0

dd∗ ∂+∂−

d∗d ∂−∂+

D2 = (d + d∗)2 = dd∗ + d∗d = −
 ∂2 = (∂+ + ∂−)2 = ∂+∂− + ∂−∂+ = −

D̃ = d−d∗ ∂̃ = ∂+ −∂−

D̃2 = (d−d∗)2 = −dd∗ −d∗d = 
 ∂̃2 = −∂+∂− −∂−∂+ = 

DD̃ = −D̃D = d∗d−dd∗ = 
̃ ∂∂̃ = −∂̃∂ = ∂−∂+ −∂+∂− = 
̃



7, 2(2005)
Differential Forms and/or Multi-vector Functions 153

defined by

Eωr =
m∑

j=0

xj ∂xj ω
r =

∑
|A|=r

dxA
m∑

j=0

xj ∂xj ω
r
A

and

EFr =
m∑

j=0

xj ∂xj Fr =
∑
|A|=r

eA

m∑
j=0

xj ∂xj Fr,A

is a scalar operator, measuring the degree of homogenicity of a function, and not af-
fecting the order of a differential form or a multi-vector function. The Euler operator
thus has the same defining expression in both worlds.

From the world of differential forms we now focus on the contraction operators
∂xj�, j = 0, 1, · · ·m, acting only on the basis elements of the differential form, but
not on the function coefficients, and given by

∂xj�dxA = ∂xj�
(
dxi1 ∧ ·· · ∧ dxir

)
=

r∑
k=1

(−1)k−1 δjik dxA\{ik } .

Apparently the contraction operator ∂xj� is, up to a minus sign, nothing else but the
”inner product”-operator (dxj • ) :

∂xj� = (−dxj • ) , j = 0, 1, · · · ,m.
However bear in mind that contractions are more fundamental than dot products. In-
deed, they can be introduced independently of a scalar product, and their behaviour
is invariant under diffeomorphisms, which is not the case for the dot product.

For a first order operator v =
m∑

j=0

vj (x) ∂xj , vj being a scalar-valued smooth func-

tion, mostly called a vector field, one may consider the associated contraction operator

v� =
m∑

j=0

vj (x) ∂xj�

which also takes the form

v� =
m∑

j=0

vj (x)(−dxj • ).

This inspires an associated ”inflation” operator

v� =
m∑

j=0

vj (x) ∂xj� =
∑
j=0

vj (x)
(
−dxj∧

)

where the action of ∂xj� = (−dxj∧) is given by
∂xj� dxA = −dxj ∧ dxA.



154 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

So from the Euler operator E we deduce the operators

E� =
∑

j

xj ∂xj� =
∑

j

xj (−dxj • )

and
E� =

∑
j

xj (−dxj ∧ )

which are in a sense complementary to the operators d and d∗ — think of replacing
xj by dxj and dxj by xj . So the operators E� and E� must show properties similar
to those of the operators d en d∗, which they indeed do, as shown in the next lemma.

Lemma 4.1

The operators E� and E� enjoy the following fundamental properties:
(i) (E�)2 = 0
(ii) (E�)2 = 0

(iii) E� + E� = −
m∑

j=0

xj (Dxj ∨)

(iv) (E� + E�)2 = E�E� + E�E� = −|x|2

The counterparts in the Clifford setting of the operators (−dxj •) and (−dxj∧)
clearly are (−ej •) and (−ej∧). The properties of the operators

m∑
j=0

xj (−ej •) = (−x •) and
m∑

j=0

xj (−ej∧) = (−x∧)

corresponding to the ones in Lemma 4.1, are then straightforward:

(i) (−x • )(−x • ) = 0
(ii) (−x∧ )(−x∧ ) = 0

(iii) (−x • ) + (−x∧ ) = −x (Clifford product understood)
(iv) ((−x • ) + (−x∧ ))2 = (−x • )(−x∧ ) + (−x∧ )(−x • ) = −|x|2.

Note that the operators (ej •) and (ej∧), j=0,1, . . ., m, coincide with the so-
called de Witt basis of the algebra of endomorphisms on the Clifford algebra R0,m+1.
Indeed, if ej and εj, j = 0, 1, . . . ,m denote the endomorphisms, given for an arbitrary
Clifford number a, by

ej : a 
−→ eja
εj : a 
−→ εja = ãej

then the Witt basis is formed by

Fj =
1
2
(ej −εj) , F′j =

1
2
(ej + εj ), j = 0, 1, . . . ,m



7, 2(2005)
Differential Forms and/or Multi-vector Functions 155

and apparently Fj = (ej •) and F′j = (ej∧).

In the same order of ideas and starting from the operators d and d∗, we introduce
the contraction and ”inflation” operators

d� =
m∑

j=0

(dxj∧) ∂xj� =
m∑

j=0

(dxj∧)(−dxj •)

d∗� =
m∑

j=0

(dxj •) ∂xj� =
m∑

j=0

(dxj •)(−dxj∧)

The operators d� and d∗� have Er(Ω) as an eigenspace since
d�ωr = r ωr and d∗�ωr = (m + 1 −r) ωr.

In other words: they measure the order of a differential form. They are sometimes
called fermionic Euler operators.
In the Clifford analysis setting we get

∂+� =
m∑

j=0

(ej∧)(−ej •) and ∂−� =
m∑

j=0

(ej •)(−ej∧)

for which indeed:

∂+� Fr = r Fr and ∂−� Fr = (m + 1 −r) Fr.
Now we turn our attention, still in the world of differential forms, to a so-called
Lie-derivative of differential forms. For a given scalar vector field v =

∑
j

vj∂xj we

define
Lvω = d v� ω + v�d ω.

It is clear that the operators Lv and d, as well as Lv and v�, commute, since
d Lv = d v� d = Lv d

and
v�Lv = v� d v� = Lv v�.

This implies that closedness and exactness of differential forms are preserved under
”Lie-derivation”.
We now prove a fundamental formula about the Lie-derivative of the Euler operator.

Lemma 4.2

For any smooth differential form ω ∈ ∧(Ω) one has
LE ω = (E�d + d E�) ω = (E + d�) ω.



156 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

Proof.

First we have

E� d ω =
∑

j

xj (−dxj •)
(∑

k

(dxk∧)∂xkω
)

=
∑

j

∑
k

xj δjk ∂xk ω +
∑

j

∑
k

xj dx
k ∧ ∂xk dxj • ω

=
∑

j

xj ∂xj +
∑

j

∑
k

xj dx
k ∧ ∂xk (dxj • ω)

while

d E� ω =
∑

k

(dxk ∧) ∂xk
∑

j

xj (−dxj • ω)

=
∑

j

∑
k

dxk ∧ (−dxj • ω) δjk −
∑

j

∑
k

dxk ∧ xj (dxj • ∂xk ω)

= −
∑

j

dxj ∧ (dxj • ω) −
∑

j

∑
k

xj dx
k ∧ ∂xk (dxj • ω).

Hence
E� d ω = E ω +

∑
j

∑
k

xj dx
k ∧ ∂xk (dxj • ω)

while
d E� ω = d� ω −

∑
j

∑
k

xj dx
k ∧ ∂xk (dxj • ω)

and the desired result follows. �

By transposing the identity of Lemma 4.1 into Clifford analysis language we get

Corollary 4.3.

For any smooth multi-vector function F ∈E(Ω) one has
(

(−x •) ∂+ + ∂+(−x •)
)
F = (E + ∂+�) F.

Corollary 4.4.

(i) For ωr ∈ ∧r(Ω) one has
LE ωr = (E� d + d E�) ωr = (E + r) ωr.

(ii) For Fr ∈Er(Ω) one has
(

(−x •) ∂+ + ∂+(−x •)
)
Fr = (E + r) Fr.



7, 2(2005)
Differential Forms and/or Multi-vector Functions 157

Corollary 4.5.

(i) If ωrk ∈
∧r(Ω) is homogeneous of degree k, then

LE ωrk = (E� d + d E�) ωrk = (k + r) ωr.
(ii) If Fr,k ∈Er (Ω) is homogeneous of degree k, then(

(−x •) ∂+ + ∂+(−x •)
)
Fr,k = (k + r) Fr,k.

The similar fundamental identity involving the operators E� and d∗ is now proven
in the language of multi-vector functions.

Corollary 4.6.

For any smooth multi-vector function F ∈E(Ω) one has
(

(−x ∧) ∂− + ∂−(−x ∧)
)
F = (E + ∂−�) F.

Proof.

On the one hand we have

x ∧ (∂−F) =
∑

j

xj (ej ∧)
∑

k

(ek •) ∂xkF

=
∑

j

xj (ej∧)(ej •) ∂xj F +
∑
j �=k

xj (ej∧)(ek •) ∂xkF

= −
∑

j

xj ∂xj
(j)

F +
∑
j �=k

xj (ej∧)(ek •) ∂xkF

where
(j)

F denotes that part of F containing the basis vector ej .
On the other hand we have

∂−(x∧F) =
∑

k

(ek •) ∂xk
∑

j

xj ej ∧F

=
∑

j

(ej •)(ej∧)F +
∑

j

∑
k

xj (ek •)(ej∧) ∂xkF

= −∂−� F +
∑

j

xj (ej •)(ej∧) ∂xj F +
∑
j �=k

xj (ek •)(ej∧) ∂xkF

= −∂−� F −
∑

j

xj ∂xj
co(j)

F +
∑
j �=k

xj (ek •)(ej∧) ∂xkF



158 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

where
co(j)

F denotes that part of F not containing the basis vector ej .
Adding both expressions yields the desired result. �

Corollary 4.7.

For any smooth differential form ω ∈ ∧(Ω) one has
(E� d∗ + d∗ E�) ω = (E + d∗�) ω.

Corollary 4.8.

(i) For ωr ∈ ∧r(Ω) one has
(E� d∗ + d∗E�) ωr = (E + m + 1 −r) ωr.

(ii) For Fr ∈Er(Ω) one has
(

(−x ∧) ∂− + ∂− (−x∧)
)
Fr = (E + m + 1 −r) Fr.

Corollary 4.9.

(i) If ωrk ∈
∧

(Ω) is homogeneous of degree k, then

(E� d∗ + d∗E�) ωrk = (k + m + 1 −r) ωrk.
(ii) If Fr,k ∈Er (Ω) is homogeneous of degree k, then

(
(−x ∧) ∂− + ∂− (−x ∧)

)
Fr,k = (k + m + 1 −r) Fr,k.

The above considerations lead to the completion of our identification table set up
at the beginning of this section.

E =
∑

j

xj ∂xj E =
∑

j

xj ∂xj

∂xj� = −dxj • −ej •

∂xj� = −dxj • −ej ∧

E� =
∑

j

xj (−dxj •)
∑

j

xj (−ej •) = −x •

E� =
∑

j

xj (−dxj ∧)
∑

j

xj (−ej ∧) = −x∧



7, 2(2005)
Differential Forms and/or Multi-vector Functions 159

E� + E� =
∑

j

xj (dx
j ∨) (−x •) + (−x∧) = −x

Clifford product understood

d� =
∑

j

(dxj ∧)(−dxj •) ∂+� =
∑

j

(ej ∧)(−ej •)

d∗� =
∑

j

(dxj •)(−dxj ∧) ∂−� =
∑

j

(ej •)(−ej ∧)

LE = d E� + E� d = E + d� ∂+ (−x •) + (−x •) ∂+ = E + ∂+�

L∗E = d∗ E� + E� d∗ = E + d∗� ∂− (−x∧) + (−x∧) ∂− = E + ∂−�

5 The Poincaré and the dual Poincaré Lemmata re-

visited

In this section we formulate refinements of the classical Poincaré Lemma and its dual,
both in the language of differential forms and in the one of multi-vector functions,
exploiting the identification established in the previous section.
As it appears to us that these refinements are rarely cited in the literature, we add
their proofs.
We start with a classical result, which in the language of three dimensional vector
fields is usually called the Helmholtz decomposition.

Proposition 5.1.

For each r-form ωr ∈ ∧r(Ω) (0 < r < m + 1) there exist ar+1 ∈ ∧r+1(Ω) and
br−1 ∈ ∧r−1(Ω) such that

(i) d ar+1 = 0 ;

(ii) d∗ br−1 = 0 ;

(iii) ωr = d∗ ar+1 + d br−1 .

Proposition 5.2.

For each r-vector function Fr ∈ Er(Ω) (0 < r < m + 1) there exist Ar+1 ∈ Er+1(Ω)
and Br−1 ∈Er−1(Ω) such that

(i) ∂+ Ar+1 = 0 ;

(ii) ∂− Br−1 = 0 ;



160 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

(iii) Fr = ∂− Ar+1 + ∂+ Br−1 .

Proof.

As the Laplace operator 
 : Er(Ω) −→ Er(Ω) is surjective (see e.g. [14], there
ought to exist Gr ∈Er(Ω) such that (−
) Gr = Fr or (∂− ∂+ + ∂+∂−) Gr = Fr .
Put Ar+1 = ∂+ Gr and Br−1 = ∂− Gr to obtain the desired result. �

Note that dωr = 0 iff the (r + 1)-form ar+1 in the above Helmholtz decomposition
is harmonic (in the sense of Hodge), while d∗ωr = 0 iff br−1 is harmonic. Similarly,
we have that ∂+Fr = 0 iff Ar+1 is monogenic, while ∂−Fr = 0 iff Br−1 is monogenic.
But there is more. The Poincaré Lemma and the Dual Poincaré Lemma will assert
that one of those harmonic forms ar+1 and br−1, respectively one of those monogenic
multi-vector functions Ar+1 and Br−1, is absorbed in the other remaining term.

Lemma 5.3. (Poincaré)

Let r ≥ 1 and let Ω be an open region contractible to a point. Then
r

ker d = d (
r−1
ker d∗ )

i.e. the following are equivalent:

(i) dωr = 0

(ii) there exists ωr−1 ∈ ∧r−1(Ω) such that d∗ωr−1 = 0 and ωr = dωr−1.
Lemma 5.4. (Poincaré)

Let r ≥ 1 and let Ω be an open region contractible to a point. Then
r

ker ∂+ = ∂+ (
r−1
ker ∂− )

i.e. the following are equivalent:

(i) ∂+Fr = ∂ ∧ Fr = 0
(ii) there exists Fr−1 ∈Er−1(Ω) such that ∂−Fr−1 = ∂ • Fr−1 = 0

and Fr = ∂+Fr−1 = ∂ ∧Fr−1 .
Proof.

We prove Lemma 5.3.
(i) =⇒ (ii)
From the classical Poincaré Lemma follows the existence of αr−1 ∈ ∧r−1(Ω) such
that ωr = dαr−1.
As 
 : ∧r−1(Ω) −→ ∧r−1(Ω) is surjective, there ought to exist βr−1 ∈ ∧r−1(Ω)



7, 2(2005)
Differential Forms and/or Multi-vector Functions 161

such that 
 βr−1 = αr−1.
Put

ωr−1 = αr−1 + dd∗βr−1.

Then clearly dωr−1 = dαr−1 = ωr. Moreover

d∗ωr−1 = d∗αr−1 + d∗dd∗βr−1 = d∗αr−1 + d∗ (dd∗ + d∗d) βr−1

= d∗ αr−1 −d∗
βr−1 = 0.

(ii) =⇒ (i)
Trivial. �

Lemma 5.5. (Dual Poincaré Lemma)

Let r < m + 1 and let Ω be an open region contractible to a point. Then

r

ker d∗ = d∗ (
r+1

ker d )

i.e. the following are equivalent:

(i) d∗ωr = 0

(ii) there exists ωr+1 ∈ ∧r+1(Ω) such that dωr+1 = 0 and ωr = d∗ωr+1.
Lemma 5.6. (Dual Poincaré Lemma)

Let r < m + 1 and let Ω be an open region contractible to a point. Then

r

ker ∂− = ∂− (
r+1

ker ∂+ )

i.e. the following are equivalent:

(i) ∂−Fr = ∂ • Fr = 0

(ii) there exists Fr+1 ∈Er+1(Ω) such that ∂+Fr+1 = 0 and Fr = ∂− Fr+1.
Proof.

We prove Lemma 5.6.
(i) =⇒ (ii)
For each Fr ∈ Er(Ω), Fr eM = Fr eoe1 . . .em = Gm+1−r belongs to Em+1−r (Ω). As
∂ Gm+1−r = (∂ Fr ) eM , we get:

∂− Gm+1−r = ∂ • Gm+1−r = [∂ Gm+1−r]m−r = [∂ Fr]r+1 eM = (∂
+ Fr) eM

and also

∂+ Gm+1−r = ∂ ∧ Gm+1−r = [∂ Gm+1−r]m+2−r = [∂ Fr ]r−1 eM = (∂− Fr) eM.



162 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

Hence Fr will satisfy ∂− Fr = 0 iff ∂+ Gm+1−r = 0. Lemma 5.4 then asserts the
existence of Gm−r ∈Em−r (Ω) such that ∂− Gm−r = 0 and Gm+1−r = ∂+ Gm−r. As
e2M = εM, εM = ± 1, we get, putting Gm−r eM εM = Fr+1 :

Fr = Gm+1−r eM εM = (∂+ Gm−r) eM εM = [∂ Gm−r]m+1−r eM εM
= [∂ Fr+1]r = ∂− Fr+1

while
∂+ Fr+1 = ∂

− Gm−r = 0.

(ii) =⇒ (i)
Trivial. �

Corollary 5.7.

If the open region Ω is contractible to a point, then the differential operators:

(i) ∂−∂+ :
r

ker ∂− −→
r

ker ∂−

(ii) ∂+∂− :
r

ker ∂+ −→
r

ker ∂+

(iii)
∼

 : Er(Ω) −→Er(Ω)

are surjective.

Proof.

(i) Take Fr ∈
r

ker ∂−. By Lemma 5.6 there exists Fr+1 ∈Er+1 Ω such that ∂+ Fr+1 =
0 and ∂− Fr+1 = Fr. So by Lemma 5.4 there exists Gr ∈Er(Ω) such that ∂−Gr = 0
and ∂+ Gr = Fr+1.

It follows that ∂−∂+ Gr = ∂− Fr+1 = Fr with Gr ∈
r

ker ∂−.

(ii) Similar to the proof of (i).

(iii) Take Fr ∈ Er (Ω). By Proposition 5.2 there exist Ar+1 ∈ Er+1(Ω) and
Br−1 ∈ Er−1(Ω) such that ∂+Ar+1 = 0, ∂−Br−1 = 0 and Fr = ∂−Ar+1 + ∂+Br−1.
By (i) and (ii) there exist Gr ∈

r

ker ∂− and Hr ∈
r

ker ∂+ such that ∂−∂+ Gr =

∂−Ar+1 ∈
r

ker ∂− and ∂+∂− Hr = −∂+Br−1 ∈
r

ker ∂+. Hence ∂−∂+(Gr + Hr) =

∂−Ar+1 and ∂+∂−(Gr + Hr) = −∂+Br−1, and thus also
∼

(Gr + Hr) = (∂−∂+ −

∂+∂−)(Gr + Hr) = ∂−Ar+1 + ∂+Br−1 = Fr. �

Now combining the Poincaré Lemma and the Dual Poincaré Lemma, we obtain
the following structure theorem on monogenic multi-vector functions and its counter-
part on harmonic differential forms.



7, 2(2005)
Differential Forms and/or Multi-vector Functions 163

Theorem 5.8.

If the open region Ω is contractible to a point, then for each ωr ∈ ∧r(Ω)
(0 < r < m + 1) the following are equivalent:

(i) ωr is harmonic in Ω, i.e. Dωr = (d + d∗) ωr = 0 in Ω

(ii) there exists ωr−1 ∈ ∧r−1(Ω) such that d∗ ωr−1 = 0, 
 ωr−1 = 0 and
ωr = dωr−1

(ii’) there exists ωr−1 ∈ ∧r−1(Ω) such that d∗ ωr−1 = 0, ∼
 ωr−1 = 0 and
ωr = dωr−1

(iii) there exists ωr+1 ∈ ∧r+1(Ω) such that d ωr+1 = 0, 
 ωr+1 = 0 and
ωr = d∗ωr+1

(iii’) there exists ωr+1 ∈ ∧r+1(Ω) such that d ωr+1 = 0, ∼
 ωr+1 = 0 and
ωr = d∗ωr+1.

Theorem 5.9.

If the open region Ω is contractible to a point, then for each Fr ∈Er(Ω)
(0 < r < m + 1) the following are equivalent:

(i) Fr is monogenic in Ω, i.e. ∂ Fr = (∂+ + ∂−) Fr = 0 in Ω

(ii) there exists Fr−1 ∈Er−1(Ω) such that ∂−Fr−1 = 0, 
 Fr−1 = 0 and
Fr = ∂+Fr−1

(ii’) there exists Fr−1 ∈Er−1(Ω) such that ∂−Fr−1 = 0,
∼

 Fr−1 = 0 and

Fr = ∂+Fr−1

(iii) there exists Fr+1 ∈Er+1(Ω) such that ∂+Fr+1 = 0, 
 Fr+1 = 0 and
Fr = ∂−Fr+1

(iii’) there exists Fr+1 ∈Er+1(Ω) such that ∂+Fr+1 = 0,
∼

 Fr+1 = 0 and

Fr = ∂−Fr+1.

Proof.

(ii) ⇒ (i) and (ii′) ⇒ (i) : trivial
(iii) ⇒ (i) and (iii′) ⇒ (i) : trivial

(i) ⇒ (ii) and (i) ⇒ (ii′)
If Fr is monogenic in Ω then ∂+Fr = 0 and ∂−Fr = 0 in Ω. By Lemma 5.4 there
exists Fr−1 ∈Er−1(Ω) such that ∂− Fr−1 = 0 and ∂+ Fr−1 = Fr. It follows that in Ω:

∂ Fr−1 = ∂
+ Fr+1 = Fr



164 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

and
(−
) Fr−1 = ∂ (∂ Fr−1) = ∂ Fr = 0.

It also follows that in Ω
∼
∂ Fr−1 = (∂

+ −∂−) Fr−1 = Fr
and ∼


 Fr−1 = ∂ (
∼
∂ Fr−1) = ∂ Fr = 0.

(i) ⇒ (iii) and (i) ⇒ (iii′)
By Lemma 5.6 there exists Fr+1 ∈Er+1 (Ω) such that ∂+ Fr+1 = 0
and ∂− Fr+1 = Fr. It follows that in Ω :

∂ Fr+1 = ∂
− Fr+1 = Fr

and
(−
) Fr+1 = ∂ (∂ Fr+1) = ∂ Fr = 0.

It also follows that in Ω
∼
∂ Fr+1 = (∂

+ −∂−) Fr+1 = −∂− Fr+1 = −Fr
and ∼


 Fr+1 = ∂ (
∼
∂ Fr+1) = −∂ Fr = 0.

�
Remarks 5.10.

(i) The above Theorems 5.8. and 5.9 may be rephrased as follows.
If the open region Ω is contractible to a point and 0 < r < m + 1, then

r

ker D = d
(

r−1
ker d∗ ∩

r−1
ker (d∗d)

)
= d (

r−1
ker 
 ∩

r−1
ker d∗)

= d (
r−1
ker

∼

 ∩

r−1
ker d∗)

r

ker D = d∗
(

r+1

ker d ∩
r+1

ker (dd∗)
)

= d∗ (
r+1

ker 
 ∩
r+1

ker d)

= d∗ (
r+1

ker
∼

 ∩

r+1

ker d)

r

ker ∂ = ∂+
(

r−1
ker ∂− ∩

r−1
ker (∂−∂+)

)
= ∂+ (

r−1
ker 
 ∩

r−1
ker ∂−)

= ∂+ (
r−1
ker

∼

 ∩

r−1
ker ∂−)

r

ker ∂ = ∂−
(

r+1

ker ∂+ ∩
r+1

ker (∂+∂−)
)

= ∂− (
r+1

ker 
 ∩
r+1

ker ∂+)

= ∂− (
r+1

ker
∼

 ∩

r+1

ker ∂+).



7, 2(2005)
Differential Forms and/or Multi-vector Functions 165

(ii) For the equivalence (i) ⇐⇒ (ii) of Theorem 5.8 we also refer to [4].

6 From the Euler operator to the Poincaré Lemma

The proof of Lemma 5.3. heavily relies on the classical Poincaré Lemma. In this
section we reflect upon the proof of this classical Poincaré Lemma and we present
an alternative proof, however restricted to real-analytic differential forms in an open
ball.
The essence of the proof of the Poincaré Lemma for one-forms is easily grasped.
Indeed, one-forms may be integrated along curves and the integral of a closed one-
form from a fixed point to a variable endpoint, in a homologically trivial domain such
as a ball, only depends on this endpoint; in other words: for closed one-forms there
is a natural notion of primitive.
For higher-order forms the integral operators in the proof of the Poincaré Lemma, are
still one-dimensional. How is it possible that such a kind of method is still successful?
The answer to this question, at least for the case of a ball, lies in considering the
Euler operator E (see also section 4).
Let P be the algebra of polynomials generated by {x0,x1, . . . ,xm} and let Pk be the
subspace of homogeneous polynomials of degree k, k ∈ N. Then it is clear
that

P =
+∞∑
k=0

⊕ Pk

is the eigenspace decomposition of P associated with the Euler operator E.
Next consider the algebra Φ of polynomial differential forms, i.e. the free associative
algebra generated by {x0,x1, . . . ,xm,dx0,dx1, . . . ,dxm}. If Φrk denotes the subspace
of r-forms with function coefficients in Pk, then one has the decomposition

Φ =
m+1∑
r=1

+∞∑
k=0

⊕ Φrk

and the question arises with which operator this decomposition is associated. The
answer to this question is given by Corollary 4.5.(i): for each ϕrk ∈ Φrk we indeed have

LE ϕrk = (E�d + dE�) ϕrk = (k + r) ϕrk ,

showing that Φrk is an eigenspace of the operator LE , which, for r ≥ 1, has only
positive eigenvalues.
The injective linear operator LE : Φ −→ Φ thus has a left inverse L−1E , given by

L−1E ϕ =
m+1∑
r=1

∑
k

L−1E (ϕrk) =
m+1∑
r=1

∑
k

1
k + r

ϕrk ,



166 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

which is also a right inverse:

L−1E LE ϕ = LE L−1E ϕ = ϕ, for all ϕ ∈ Φ.
Moreover, as in the case for LE , the operator L−1E commutes with the operators d
and E� :

L−1E d = dL−1E and L−1E E� = E� L−1E .
For any polynomial differential form ϕ, not containing a scalar part, we thus have

ϕ = L−1E E�d ϕ + L−1E d E�ϕ

= L−1E E�d ϕ + d L−1E E�ϕ
and, in particular, for any closed polynomial differential form ϕclosed we find

ϕclosed = d (L−1E E� ϕclosed) = d (E�L−1E ϕclosed).
This proves the Poincaré Lemma for closed polynomial differential forms in any open
region of Rm+1.
Finally, let ωr be a closed real-analytic r-form in a ball centred at the origin, say
◦
B(0,R). Then the series

ωr(x) =
∞∑

k=0

ωrk (x) , ω
r
k ∈ Φrk

together with all its derived series, converges uniformly on the compact subsets of
◦
B(0,R). As for each k,

L−1E ωrk =
1

k + r
ωrk

and as the series ∞∑
k=0

1
k + r

ωrk (x)

together with all its derived series, also converges uniformly on the compact subsets

of
◦
B(0,R), we may define

L−1E ωr =
∞∑

k=0

L−1E ωrk.

Hence

ωr =
∞∑

k=0

ωrk =
∞∑

k=0

d (E� L−1E ωrk) = d (E�
∞∑

k=0

L−1E ωrk)

= d (E� L−1E ωr)
which concludes the proof of the Poincaré Lemma for closed real-analytic r-forms in
an open ball centred at the origin.



7, 2(2005)
Differential Forms and/or Multi-vector Functions 167

Remark 6.1.

In a similar way the Dual Poincaré Lemma for co-closed real-analytic differential
forms in an open ball may be proved. The key steps in the proof are

(i) Corollary 4.8.(i) stating that for each ϕrk ∈ Φrk :

L∗E ϕrk = (E�d∗ + d∗E�) ϕrk = (k + m + 1 −r) Erk ;

(ii) the commutation rules:

d∗L∗E = d∗E�d∗ = L∗E d∗
E�L∗E = E�d∗E� = L∗E E� ;

(iii) the inversion formula for a polynomial differential form ϕ :

ϕ = (L∗−1E E�d∗ + L∗−1E d∗E�) ϕ ;

(iv) and in particular for a co-closed polynomial differential form ϕco−closed :

ϕco−closed = L∗−1E d∗E�ϕco−closed = d∗ (L∗−1E E�) ϕco−closed.

7 Differential forms versus multivector functions

In the previous sections we established and illustrated a ”natural” isomorphism be-
tween on the one hand the Cartan algebra of differential forms (extended with the
Hodge star operator and the inner product or dot product), with the underlying
structure of the Grassmann algebra, and on the other hand the algebra of multi-
vector functions in Clifford analysis with the underlying structure of Clifford algebra.
This could easily lead to the conclusion that either one of both is redundant. In-
deed it is true that the equations of Clifford analysis may often be rewritten using
vector calculus or more generally differential forms. This is nicely illustrated by the
correspondence table of section 4 and in particular by the correspondence between
the action of the Dirac operator ∂ on multi-vector functions and the action of the
operator D = d + d∗ on differential forms. Historically this redundancy issue has led
to a long and repeated discussion between those who advocate the use of differential
forms and those who consider differential forms as an intermediate concept that can
be fully replaced by Clifford algebra. Examples of papers where Clifford algebra is
realized by means of Grassmann algebra are [7], [9] and [12].
A typical construct in these is the so-called ”vee-product” or Clifford product of dif-
ferential forms (see e.g. [2]). The Dirac operator D on the Cartan algebra

∧
(Ω) may

then be defined by
Dω = D ∨ω , ω ∈

∧
(Ω) .



168 F. Brackx, R. Delanghe and F. Sommen
7, 2(2005)

It turns out that D = d + d∗.
On the other hand, in their book [6] Hestenes and Sobczyk recover most of the theory
and calculus of differential forms by interpreting them as alternating tensors which
may be represented by means of linear functions on the subspaces of r-vectors in a
Clifford algebra, an approach which was made more explicit in [5].
Strictly speaking both points of view are mathematically correct. What we do not
agree with is the conclusion that either the use of an extra Clifford basis (e0,e1, . . . ,em)
next to (dx0,dx1, . . . ,dxm) or the use of the differential forms
dx0,dx1, . . . ,dxm as basic elements of calculus, is redundant. Despite the similarities
as depicted in this paper, the dxj and ej are different calculus objects with a different
calculus behaviour, which will be fully demonstrated and illustrated in the forthcom-
ing paper [13].
Many examples illustrating the falsity of the ”redundancy idea” could be given, but
the main counter-argument relies in the success and the richness of the results ob-
tained by considering both the basic differential forms dxj and the Clifford algebra
generators ej as independent calculus elements. This is nicely demonstrated in e.g.
[11] where Chapter 9 focusses on the interplay between complex differential forms and
complex Clifford algebras and its usefulness for classical several complex variables
theory is shown.

Received: June 2004. Revised: August 2004.

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