Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 23, 2006, 71–93 \begin{matrix}c \\ ◯ \end{matrix} D. Krupka

D. Krupka

THE TOTAL DIVERGENCE EQUATION

(submitted by V.V. Lychagin)

Abstract. In this paper, the total divergence equation is investigated by means of the methods used in the theory of finite order variational sequences. Integrability conditions for this equation are found, and all solutions are described. The correspondence of the solutions with some differential forms on jet spaces is established.

1. Introduction

Let U ⊂ ℝ^{n} be an open set, let W ⊂ ℝ^{m} be an open ball with center at the origin, and denote V = U × W. We consider V as a fibered manifold over U with the first Cartesian projection π : V → U. V^{s} denotes the s-jet prolongation of V; explicitly, V^{s} = U × W × L(ℝ^{n}, ℝ^{m}) × L_{sym}^{2}(ℝ^{n}, ℝ^{m})×...×L_{sym}^{s}(ℝ^{n}, ℝ^{m}), where L_{sym}^{k}(ℝ^{n}, ℝ^{m}) is the vector space

2000 Mathematical Subject Classification. primary: 35F20, secondary: 58A15, 58E30.

Key words and phrases. Differential equation, jet, divergence, integrability, variational sequence, Euler-Lagrange expression.

The author is grateful to the Mathematics Department and the Institute for Advanced Study at La Trobe University, Melbourne, where as an IAS Distinguished Fellow he worked on the research project Differential Equations and Their Variational Properties. The author also appreciates the support of the Czech Science Foundation (grant 201/06/0922), and the Czech Ministry of Education, Youth, and Sports (grant MSM 6198959214).

of k-linear, symmetric mappings from ℝ^{n} to ℝ^{m} . The Cartesian coordinates on V, and the associated jet coordinates on V^{s} , are denoted by x^{i}, y^{σ}, and x^{i}, y^{σ}, y_{j_{1}}^{σ}, y_{j_{1}j_{2}}^{σ}, ..., y_{j_{1}j_{2}...j_{s}}^{σ}, respectively; in these expressions 1 ≤ i, j_{1}, j_{2}, ..., j_{s} ≤ n and 1 ≤ σ ≤ m.

Let f : V^{r} → ℝ be a differentiable function (order of differentiability in different variables can be easily deduced from the context). Our aim in this paper is to find solutions g = (g^{1}, g^{2}, ..., g^{n}) of the total divergence equation

d_{i}g^{i} = f,  (1)

whose components g^{i} are differentiable real-valued functions on V^{s}, where s is a positive integer. Since the total divergence d_{i}g^{i} is defined by

d_{i}g^{i} = \frac{∂g^{i}}{∂x^{i}} + \frac{∂g^{i}}{∂y^{σ}} y_{i}^{σ} + \frac{∂g^{i}}{∂y_{i_{1}}^{σ}} y_{i_{1}i}^{σ} + \frac{∂g^{i}}{∂y_{i_{1}i_{2}}^{σ}} y_{i_{1}i_{2}i}^{σ} + ... + \frac{∂g^{i}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} y_{i_{1}i_{2}...i_{r}i}^{σ}, (2)

equation (1) is a first order partial differential equation. From this expression we immediately see that every solution g = g^{i} , defined on V^{s} , such that s ≤ r + 1, satisfies

\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{s+1}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}i_{4}...i_{s+1}}^{σ}} + ... + \frac{∂g^{i_{s}}}{∂y_{i_{1}i_{2}...i_{s−1}i_{s+1}}^{σ}} + \frac{∂g^{i_{s+1}}}{∂y_{i_{1}i_{2}...i_{s}}^{σ}} = 0. (3)

In this paper, we solve the total divergence equation by means of the methods, developed in the theory of finite order variational sequences (Krupka [12]). For additional information on this theory, see also e.g. Francaviglia, Palese, Vitolo [3], Grassi [4], Grigore [5], Krbek and Musilova [9], Pommaret [14], and Vitolo [19]. We use standard concepts of the calculus of differential forms on jet prolongations of fibered manifolds, applied to open sets in Euclidean topological spaces ℝ^{n} (see e.g. Krupka [10], [11]). A systematic exposition of variational analysis and differential equations in this context can be found in Hakova and Krupkova [6], and Krupkova [13]. We prove two theorems on the structure of solutions of the total divergence equation. We consider this equation as an overdetermined equation, and find the corresponding integrability condition, which guarantees existence of solutions. It turns out that integrability of the total divergence equation is equivalent with the vanishing of the Euler-Lagrange expressions of the right-hand side of equation (1), well known from the multi-dimensional, higher order variational theory. If the integrability condition is satisfied, we find all solutions. Moreover we show that the solutions can naturally be interpreted as some differential forms. The tools we use include among others the theory of Lepage forms, the Young decompositions and the trace decompositions of tensor spaces, and the fibered homotopy operator.

It seems that the total divergence equations, differential operators, and algebraic constructions related with them, appeared for the first time in the geometric theory of partial differential equations on infinite jet spaces and, in particular, in the variational bicomplex theory (see e.g. Anderson [1], Dedecker and Tulczyjew [2], Krasilschik [7], Krasilschik, Lychagin, and Vinogradov [8], Saunders [15], Takens [16], Tulczyjew [17], Vinogradov [18], Tsujishita [20], and references therein). These equations play a crucial role in the proofs of exactness of the bicomplex. However, no explicit discussion has appeared in this context yet; the structure of solutions of the total divergence equation, and the meaning of condition (3), in these sources remain unclear.

2. Notation

In what follows we use basic notions from the theory of contact forms on jet prolongations of fibered manifolds (see e.g. [11], [13]). We denote

ω_{0} = dx^{1} ∧ dx^{2} ∧ ... ∧ dx^{n}, ω_{i} = i_{∂/∂x^{i}}ω_{0}, ω_{ij} = i_{∂/∂x^{i}}i_{∂/∂x^{j}}ω_{0}.

The contact 1-forms ω_{j_{1}j_{2}...j_{k}}^{σ} are defined by

ω_{j_{1}j_{2}...j_{k}}^{σ} = dy_{j_{1}j_{2}...j_{k}}^{σ} − y_{j_{1}j_{2}...j_{k}l}^{σ}dx^{l}, k = 0, 1, 2, ..., s − 1.

h and p_{k} denote the horizontalization, and the k-contact mappings.

For any smooth function f : V^{r} → ℝ we define an n-form λ_{f} on V^{r} and a system of functions E_{σ}(f) : V^{2r} → ℝ by

λ_{f} = fω_{0},

and

E_{σ}(f) = \frac{∂f}{∂y^{σ}} − d_{i_{1}} \frac{∂f}{∂y_{i_{1}}^{σ}} + d_{i_{1}}d_{i_{2}} \frac{∂f}{∂y_{i_{1}i_{2}}^{σ}} − d_{i_{1}}d_{i_{2}}d_{i_{3}} \frac{∂f}{∂y_{i_{1}i_{2}i_{3}}^{σ}}

+... + (−1)^{r−1}d_{i_{1}}d_{i_{2}}...d_{i_{r−1}} \frac{∂f}{∂y_{i_{1}i_{2}...i_{r−1}}^{σ}} + (−1)^{r}d_{i_{1}}d_{i_{2}}...d_{i_{r}} \frac{∂f}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} .

λ_{f} is the lagrangian associated with f, and the (n + 1)-form

E_{f} = E_{σ}(f)ω^{σ} ∧ ω_{0}

is the Euler-Lagrange form, associated with f.

3. Projectable extensions of horizontal (n − 1)-forms

Let us consider a π^{r}-horizontal (n − 1)-form η on V^{r}, expressed as

η = g^{i}ω_{i} = \frac{1}{(n − 1)!}h_{j_{2}j_{3}...j_{n}}dx^{j_{2}} ∧ dx^{j_{3}} ∧ ... ∧ dx^{j_{n}}.  (4)

Since

ω_{i} = \frac{1}{(n − 1)!}ε_{ij_{2}j_{3}...j_{n}}dx^{j_{2}} ∧ dx^{j_{3}} ∧ ... ∧ dx^{j_{n}},

we have the transformation formulas

h_{j_{2}j_{3}...j_{n}} = ε_{ij_{2}j_{3}...j_{n}}g^{i}, g^{k} = \frac{1}{(n − 1)!}ε^{kj_{2}j_{3}...j_{n}}h_{j_{2}j_{3}...j_{n}}.

We prove the following assertion, in which alt and sym denote alternation and symmetrization in the corresponding indices.

Lemma 1. The functions g^{i} and h_{j_{1}j_{2}...j_{n−1}} satisfy

\frac{1}{(r+1)}ε_{il_{2}l_{3}...l_{n}}(\frac{∂g^{i}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} + \frac{∂g^{k_{1}}}{∂y_{ik_{2}k_{3}...k_{r}}^{σ}} + \frac{∂g^{k_{2}}}{∂y_{k_{1}ik_{3}k_{4}...k_{r}}^{σ}} +...+ \frac{∂g^{k_{r}}}{∂y_{k_{1}k_{2}...k_{r−1}i}^{σ}})

= \frac{∂h_{l_{2}l_{3}...l_{n}}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} − \frac{r(n − 1)}{(r + 1)} \frac{∂h_{il_{3}l_{4}...l_{n}}}{∂y_{ik_{2}k_{3}...k_{r}}^{σ}} δ_{l_{2}}^{k_{1}} alt(l_{2}l_{3}...l_{n}) sym(k_{1}k_{2}...k_{r}).

Proof. We have

\frac{∂g^{i}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} \frac{1}{(n − 1)!}ε^{ij_{2}j_{3}...j_{n}} \frac{∂h_{j_{2}j_{3}...j_{n}}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} .

Multiplying this expression by ε_{il_{2}l_{3}...l_{n}} we obtain

ε_{il_{2}l_{3}...l_{n}} \frac{∂g^{i}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} = \frac{1}{(n − 1)!}ε_{il_{2}l_{3}...l_{n}}ε^{ij_{2}j_{3}...jn} \frac{∂h_{j_{2}j_{3}...j_{n}}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} = \frac{∂h_{l_{2}l_{3}...l_{n}}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} .

The rest of the proof is routine. □

We say that a π^{r}-horizontal form η, defined on V^{r}, has a projectable extension, if there exists a form μ on V^{r,r−1} such that

η = hμ.

Let us consider a form η, expressed in two bases of (n − 1)-forms by (4).

Lemma 2. The following three conditions are equivalent:

(a) η has a π^{r,r−1}-projectable extension.

(b) The components h_{i_{1}i_{2}...i_{n−1}} satisfy

\frac{∂h_{i_{1}i_{2}...i_{n−1}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} −\frac{r(n − 1)}{r + 1} \frac{∂h_{si_{2}i_{3}...i_{n−1}}}{∂y_{sj_{2}j_{3}...j_{r}}^{σ}} δ_{i_{1}}^{j_{1}} = 0 sym(j_{1}j_{2}...j_{r}) alt(i_{1}i_{2}...i_{n−1}).

(c) The components g^{i} satisfy

\frac{∂g^{i}}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} + \frac{∂g^{k_{1}}}{∂y_{ik_{2}k_{3}...k_{r}}^{σ}} + \frac{∂g^{k_{2}}}{∂y_{k_{1}ik_{3}k_{4}...k_{r}}^{σ}} + ... + \frac{∂g^{k_{r}}}{∂y_{k_{1}k_{2}...k_{r−1}i}^{σ}} = 0.

Proof. 1. To show that (a) implies (b), suppose that we have an (n − 1)- form μ on V^{r−1} such that η = hμ. To express μ in a fibered chart, we use multi-indices J = (j_{1}j_{2}...j_{r−1}) of length |J| ≤ r − 1. Then (π^{r,r−1})^{∗}μ is expressible as

(π^{r,r−1})^{∗}μ = \frac{1}{(n − 1)!}h_{i_{1}i_{2}...i_{n−1}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{n−1}}

+ ∑_{k=1}^{n−2} \frac{1}{k!(n − 1 − k)!}h_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{k}i_{k+1}i_{k+2}...i_{n−1}}^{J_{k}}ω_{J_{1}}^{σ_{1}} ∧ ω_{J_{2}}^{σ_{2}} ∧ ... ∧ ω_{J_{k}}^{σ_{k}}

∧dx^{i_{k+1}} ∧ dx^{i_{k+2}} ∧ ... ∧ dx^{i_{n−1}}

+\frac{1}{(n − 1)!}h_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{n−1}}^{J_{n−1}}ω_{J_{1}}^{σ_{1}} ∧ ω_{J_{2}}^{σ_{2}} ∧ ... ∧ ω_{J_{n−1}}^{σ_{n−1}}.

Since μ is defined on V^{r−1}, so is dμ. In particular, all terms in (π^{r,r−1})^{∗}dμ = d(π^{r,r−1})^{∗}μ, containing dy_{j_{1}j_{2}...j_{r}}^{ν}, should vanish identically. We find the terms in d(π^{r,r−1})^{∗}μ, which do not contain any of the forms ω^{σ}, ω_{j_{1}}^{σ}, ω_{j_{1}j_{2}}^{σ}, ..., ω_{j_{1}j_{2}...j_{r−1}}^{σ}; these terms should vanish separately. Obviously, they can arise only from the summands

\begin{matrix}\frac{1}{(n − 1)!}h_{i_{1}i_{2}...i_{n−1}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{n−1}} \\ +\frac{1}{(n − 2)!}h_{σ_{1}i_{2}i_{3}...i_{n−1}}^{J_{1}}ω_{J_{1}}^{σ_{1}} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ ... ∧ dx^{i_{n−1}} \end{matrix}  (5)

in η, in which |J_{1}| = r − 1. Writing J_{1} = (j_{1}j_{2}...j_{r−1}), differentiating (5) and omitting the terms containing ω_{J_{1}}^{σ_{1}}, we get

\frac{1}{(n − 1)!}d′_{i_{1}}h_{i_{2}i_{3}...i_{n}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ ... ∧ dx^{i_{n}}

+\frac{1}{(n − 1)!} \frac{∂h_{i_{2}i_{3}...i_{n}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} dy_{j_{1}j_{2}...j_{r}}^{σ} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ ... ∧ dx^{i_{n}}

−\frac{1}{(n − 2)!}h_{σ}^{j_{1}j_{2}...j_{r−1}}_{i_{3}i_{4}...i_{n}}dy_{j_{1}j_{2}...j_{r−1}i_{2}}^{σ} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ dx^{i_{4}} ∧ ... ∧ dx^{i_{n}}

= \frac{1}{(n − 1)!}d′_{i_{1}}h_{i_{2}i_{3}...i_{n}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ ... ∧ dx^{i_{n}}

+\frac{1}{(n − 2)!} (\frac{1}{n − 1} \frac{∂h_{i_{2}i_{3}...i_{n}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} − h_{σ}^{j_{1}j_{2}...j_{r−1}}_{i_{3}i_{4}...i_{n}}δ_{i_{2}}^{j_{r}})

·dy_{j_{1}j_{2}...j_{r}}^{σ} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ ... ∧ dx^{i_{n}}, where d′_{i} denotes the cut formal derivative,

d′_{i_{1}}h_{i_{2}i_{3}...i_{n}} = d_{i_{1}}h_{i_{2}i_{3}...i_{n}} − \frac{∂h_{i_{2}i_{3}...i_{n}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} y_{j_{1}j_{2}...j_{r}i_{1}}^{σ}.

Consequently, d(π^{r,r−1})^{∗}μ can be written as

(π^{r,r−1})^{∗}dμ = \frac{1}{(n − 1)!}d′_{i_{1}}h_{i_{2}i_{3}...i_{n}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ ... ∧ dx^{i_{n}}

+\frac{1}{(n − 2)!} (\frac{1}{n − 1} \frac{∂h_{i_{2}i_{3}...i_{n}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} − h_{σ}^{j_{1}j_{2}...j_{r−1}}_{i_{3}i_{4}...i_{n}}δ_{i_{2}}^{j_{r}})

·dy_{j_{1}j_{2}...j_{r}}^{σ} ∧ dx^{i_{2}} ∧ dx^{i_{3}} ∧ ... ∧ dx^{i_{n}} + τ,

where τ is generated by the forms ω^{σ}, ω_{j_{1}}^{σ}, ω_{j_{1}j_{2}}^{σ}, ..., ω_{j_{1}j_{2}...j_{r−1}}^{σ}. But d(π^{r,r−1})^{∗}μ is π^{r,r−1}-projectable, so we see that μ satisfies

\begin{matrix}\frac{1}{n − 1} \frac{∂h_{i_{1}i_{2}...i_{n−1}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} − h_{σ}^{j_{1}j_{2}...j_{r−1}}_{i_{2}i_{3}...i_{n−1}}δ_{i_{1}}^{j_{r}} = 0 \\ alt(i_{1}i_{2}...i_{n−1}) sym(j_{1}j_{2}...j_{r}). \end{matrix}  (6)

The structure of this system of algebraic equations for h_{σ}^{j_{1}j_{2}...j_{r−1}}_{i_{3}i_{4}...i_{n}} is described by the trace decomposition theory. In what follows we use the trace operation tr , and a complementary operation, denoted by q , which satisfy U = tr q U + q tr U for any tensor U, symmetric in the superscripts, and antisymmetric in the subscripts (see [11]). In terms of these operations, equation (6) can be written in the form q X = A, where

X = h_{σ}^{j_{1}j_{2}...j_{r−1}}_{i_{1}i_{2}...i_{n−2}}, A = \frac{r}{r + 1} \frac{∂h_{i_{1}i_{2}...i_{n−1}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} .

Recall that

q X = \frac{r(n − 1)}{r + 1}h_{σ}^{j_{2}j_{3}...j_{r}}_{i_{2}i_{3}...i_{n−1}}δ_{i_{1}}^{j_{1}} alt(i_{1}i_{2}...i_{n−1}) sym(j_{1}j_{2}...j_{r}).

But we have the identities A = tr q A + q tr A and q q X = 0, and we know that a necessary and sufficient condition for existence of a solution X is q A = 0 or, equivalently, A − q tr A = 0. Since

q tr A = \frac{r^{2}(n − 1)}{(r + 1)^{2}} \frac{∂h_{si_{2}i_{3}...i_{n−1}}}{∂y_{sj_{2}j_{3}...j_{r}}^{σ}} δ_{i_{1}}^{j_{1}} sym(j_{1}j_{2}...j_{r}) alt(i_{1}i_{2}...i_{n−1}),

and

A − q tr A = \frac{r}{r + 1} \frac{∂h_{i_{1}i_{2}...i_{n−1}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} − \frac{r^{2}(n − 1)}{(r + 1)^{2}} \frac{∂h_{si_{2}i_{3}...i_{n−1}}}{∂y_{sj_{2}j_{3}...j_{r}}^{σ}} δ_{i_{1}}^{j_{1}}

sym(j_{1}j_{2}...j_{r}) alt(i_{1}i_{2}...i_{n−1}), we see that (a) implies (b). Note that condition (b) means that the expression

\frac{∂h_{i_{1}i_{2}...i_{n−1}}}{∂y_{j_{1}j_{2}...j_{r}}^{σ}}

has vanishing its traceless component.

2. Property (c) of the form η follows from (b) by Lemma 1.

3. Suppose that the functions g^{i} satisfy condition (c),

\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{r+1}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}...i_{r+1}}^{σ}} + \frac{∂g^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}...i_{r+1}}^{σ}} + ... + \frac{∂g^{i_{r+1}}}{∂y_{i_{2}i_{3}...i_{r}i_{1}}^{σ}} = 0. (7)

We wish to show that these functions are necessarily polynomials of degree ≤ n−1 in the variables y_{j_{1}j_{2}...j_{r}}^{ν}. In fact we prove that every solution g^{i} of equation (7) is necessarily a polynomial in the variables y_{J}^{ν} of degree ≤ n − 1. It will be convenient to work with multi-indices of length r, J = (j_{1}j_{2}...j_{r}). We assert that

\frac{∂^{n}g^{i}}{∂y_{J_{1}}^{σ_{1}}∂y_{J_{2}}^{σ_{2}}...∂y_{J_{n}}^{σ_{n}}} = 0.  (8)

We prove (8) by showing that all Young diagrams, defining the Young decomposition of the expression on the left of (8), vanish. Since this expression is symmetric in the indices entering every of the multi-indices J_{1}, J_{2}, ..., J_{n}, only the diagrams, which contain any of the blocks J_{1}, J_{2}, ..., J_{n} in a row, can make a nonzero contribution. Thus, we can restrict our attention to typical diagrams, which include the blocks J_{1}, J_{2}, ..., J_{n} as follows:

J_{1}  J_{2}  J_{3}  ...  ...  J_{k_{1}}
J_{k_{1}+1}  J_{k_{1}+2}  ...  J_{k_{1}+k_{2}}
J_{k_{1}+k_{2}+1} J_{k_{1}+k_{2}+2} ...

...

(diagrams with different position of indices in each row give analogous Young projectors). On the other hand, the Young diagrams for the decomposition of (8) should also include the index i. If this index stands in a row, which contains at least one of the blocks J_{1}, J_{2}, ..., J_{n}, we get necessarily the zero Young projector, by (7). Thus, a non-zero projector may possibly arise only from the diagram, in which i is placed on the bottom:

J_{1}  J_{2}  J_{3}  ...  ...  J_{k_{1}}
J_{k_{1}+1}  J_{k_{1}+2}  ...  J_{k_{1}+k_{2}}
J_{k_{1}+k_{2}+1} J_{k_{1}+k_{2}+2} ...

...

i

However, then it follows again from (7) and from antisymmetry of the corresponding Young projector in the first column, that we get the zero contribution whenever k_{1} ≥ 2; otherwise we simply transform the index i to the first column by a permutation in the first column. Thus, a nonzero contribution may arise only for the diagram

\begin{matrix}J_{1} \\ J_{2} \\ ... \\ J_{n} \\ i \end{matrix}

But the Young projector, corresponding with this diagram, is the zero projector because this diagram contains n + 1 rows. This proves that g^{i} satisfies condition (8). In particular, g^{i} must be a polynomial in y_{i_{1}i_{2}...i_{s}}^{σ} of degree ≤ n − 1.

Let g_{p}^{i} be the homogeneous component of g^{i} of degree p. Then

g^{i} = g_{0}^{i} + g_{1}^{i} + g_{2}^{i} + ... + g_{n−1}^{i},  (9)

and we have for each p = 0, 1, 2, ..., n − 1

\frac{∂g_{p}^{i_{1}}}{∂y_{i_{2}i_{3}...i_{r+1}}^{σ}} + \frac{∂g_{p}^{i_{2}}}{∂y_{i_{1}i_{3}...i_{r+1}}^{σ}} + \frac{∂g_{p}^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}...i_{r+1}}^{σ}} + ... + \frac{∂g_{p}^{i_{r+1}}}{∂y_{i_{2}i_{3}...i_{r}i_{1}}^{σ}} = 0.

Moreover, since for every p = 1, 2, ..., n − 1

g_{p}^{i} = \frac{1}{p} \frac{∂g_{p}^{i}}{∂y_{I_{1}}^{σ_{1}}} y_{I_{1}}^{σ_{1}},

we have

g_{p}^{i} = \frac{1}{p^{2}} \frac{∂^{2}g_{p}^{i}}{∂y_{I_{1}}^{σ_{1}}∂y_{I_{2}}^{σ_{2}}} y_{I_{1}}^{σ_{1}}y_{I_{2}}^{σ_{2}} + \frac{1}{p}g_{p}^{i},

i.e.,

g_{p}^{i} = \frac{1}{p(p − 1)} \frac{∂^{2}g_{p}^{i}}{∂y_{I_{1}}^{σ_{1}}∂y_{I_{2}}^{σ_{2}}} y_{I_{1}}^{σ_{1}}y_{I_{2}}^{σ_{2}}.

Continuing we get

g_{p}^{i} = \frac{1}{p!} \frac{∂^{p}g_{p}^{i}}{∂y_{I_{1}}^{σ_{1}}∂y_{I_{2}}^{σ_{2}}...∂y_{I_{p}}^{σ_{p}}} y_{I_{1}}^{σ_{1}}y_{I_{2}}^{σ_{2}}...y_{I_{p}}^{σ_{p}},  (10)

where the coefficients are independent of y_{I}^{σ} , |I| = r. Writing

\frac{∂^{p}g_{p}^{i}}{∂y_{I_{1}}^{σ_{1}}∂y_{I_{2}}^{σ_{2}}...∂y_{I_{p}}^{σ_{p}}} = \frac{∂^{p}g_{p}^{i}}{∂y_{J_{1}j_{1}}^{σ_{1}}∂y_{J_{2}j_{2}}^{σ_{2}}...∂y_{J_{p}j_{p}}^{σ_{p}}}  (11)

and analyzing this expression in the same way as above, we see that we have a unique Young diagram giving a possibly nontrivial Young symmetrizer, namely

j_{1} J_{1}
j_{2} J_{2}
...
j_{p−1} J_{p−1}
j_{p} J_{p}
i

Thus, the coefficients (11) are antisymmetric in i, j_{1}, j_{2}, ..., j_{p}. This completes our description of the functions g^{i} (9), satisfying condition (7).

Write for the coefficients in (10)

\frac{∂^{p}g_{p}^{i}}{∂y_{I_{1}}^{σ_{1}}∂y_{I_{2}}^{σ_{2}}...∂y_{I_{p}}^{σ_{p}}} = \frac{∂^{p}g_{p}^{j}}{∂y_{J_{1}i_{1}}^{σ_{1}}∂y_{J_{2}i_{2}}^{σ_{2}}...∂y_{J_{p}i_{p}}^{σ_{p}}} δ_{j}^{i}δ_{i_{1}}^{j_{1}}δ_{i_{2}}^{j_{2}}...δ_{i_{p}}^{j_{p}} alt(ij_{1}j_{2}...j_{p})

= \frac{1}{(p + 1)!(n − p − 1)!} \frac{∂^{p}g_{p}^{i}}{∂y_{J_{1}i_{1}}^{σ_{1}}∂y_{J_{2}i_{2}}^{σ_{2}}...∂y_{J_{p}i_{p}}^{σ_{p}}}

·ε_{ii_{1}i_{2}...i_{p−1}i_{p}l_{p+1}...l_{n−1}}ε^{jj_{1}j_{2}...j_{p−1}j_{p}l_{p+1}...l_{n−1}}

where I_{p} = J_{p}j_{p}. Set

A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{p}}^{J_{p}}_{l_{p+1}l_{p+2}...l_{n−1}} = \frac{1}{(p + 1)!} \frac{∂^{p}g_{p}^{i}}{∂y_{J_{1}k_{1}}^{σ_{1}}∂y_{J_{2}k_{2}}^{σ_{2}}...∂y_{J_{p}k_{p}}^{σ_{p}}} ε_{ik_{1}k_{2}...k_{p}l_{p+1}l_{p+2}...l_{n−1}},

and for each p = 1, 2, ..., n − 1

μ_{p} = \frac{1}{p!(n − p − 1)!}A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{p}}^{J_{p}}_{l_{p+1}l_{p+2}...l_{n−1}}dy_{J_{1}}^{σ_{1}} ∧ dy_{J_{2}}^{σ_{2}} ∧ ... ∧ dy_{J_{p}}^{σ_{p}}

∧dx^{l_{p+1}} ∧ dx^{l_{p+2}} ∧ ... ∧ dx^{l_{n−1}}.

We extend this definition for p = 0, setting

μ_{0} = g_{0}^{i}ω_{i}.

Let us now consider the form

μ = μ_{0} + μ_{1} + μ_{2} + ... + μ_{n−1}.

80 D. KRUPKA μ is obviously defined on V^{r−1}. Since

ε^{ik_{2}k_{3}...k_{n}}ω_{i} = dx^{k_{2}} ∧ dx^{k_{3}} ∧ ... ∧ dx^{k_{n}},

we have

hμ_{p} = \frac{1}{p!(n − p − 1)!}A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{p}}^{J_{p}}_{l_{p+1}l_{p+2}...l_{n−1}}y_{J_{1}i_{1}}^{σ_{1}}y_{J_{2}i_{2}}^{σ_{2}}...y_{J_{p}i_{p}}^{σ_{p}}

·dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{p}} ∧ dx^{l_{p+1}} ∧ ... ∧ dx^{l_{n−1}}

= (p + 1)\frac{1}{(p + 1)!} \frac{∂^{p}g_{p}^{i}}{∂y_{J_{1}i_{1}}^{σ_{1}}∂y_{J_{2}i_{2}}^{σ_{2}}...∂y_{J_{p}i_{p}}^{σ_{p}}} y_{J_{1}i_{1}}^{σ_{1}}y_{J_{2}i_{2}}^{σ_{2}}...y_{J_{p}i_{p}}^{σ_{p}}ω_{i}

= g_{p}^{i}ω_{i},

in accordance with (10). Now we can conclude that

hμ = hμ_{0} + ∑_{p=1}^{n−1} hμ_{p} = g_{0}^{i}ω_{i} + ∑_{p=1}^{n−1} g_{p}^{i}ω_{i} = g^{i}ω_{i} = η.

This completes the proof of Lemma 2. □

4. Lepage equivalents of a lagrangian

We have the following assertion. Lemma 3. For any function f : V^{r} → ℝ, there exists an n-form Θ, defined on V^{2r−1}, such that (a) λ = hΘ, and (b) the form p_{1}dΘ is ω^{σ}- generated. Proof. We search for Θ of the form

Θ=fω_{0}+(f^{i}_{σ}ω^{σ}+f^{i}_{σ}^{j_{1}}ω_{j_{1}}^{σ}+f^{i}_{σ}^{j_{1}j_{2}}ω_{j_{1}j_{2}}^{σ}+...+f^{i}_{σ}^{j_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ}) ∧ ω_{i}, (12)

with undetermined coefficients f^{i}_{σ}, f^{i}_{σ}^{j_{1}}, f^{i}_{σ}^{j_{1}j_{2}}, ..., f^{i}_{σ}^{j_{1}j_{2}...j_{r−1}}. Note that

\begin{matrix}ω_{j_{1}j_{2}...j_{r−1}}^{σ_{1}} ∧ ω_{i} = ω_{j_{1}j_{2}...j_{r−1}}^{σ_{1}} ∧ ω_{i} sym(j_{1}j_{2}...j_{r−1}i) \\ +\frac{r − 1}{r}d(ω_{j_{2}j_{3}...j_{r−1}}^{σ_{1}} ∧ ω_{j_{1}i}) sym(j_{1}j_{2}...j_{r−1}). \end{matrix}  (13)

Denote by f̃_{σ}^{ij_{1}j_{2}...j_{r−1}} the expression f^{i}_{σ}^{j_{1}j_{2}...j_{r−1}}, symmetrized in the superscripts. Applying formula (13) to the last term in (12), we obtain

f^{i}_{σ}^{j_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ} ∧ ω_{i} = f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ} ∧ ω_{i}

+\frac{r − 1}{r}f^{i}_{σ}^{j_{1}j_{2}...j_{r−1}}d(ω_{j_{2}j_{3}...j_{r−1}}^{σ} ∧ ω_{j_{1}i})

= f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ} ∧ ω_{i}

+\frac{r − 1}{r}d_{k}(f^{k}_{σ}^{ij_{2}j_{3}...j_{r−1}} − f^{i}_{σ}^{kj_{2}j_{3}...j_{r−1}})ω_{j_{2}j_{3}...j_{r−1}}^{σ} ∧ ω_{i}

+\frac{r − 1}{r}d(f^{i}_{σ}^{j_{1}j_{2}...j_{r−1}}ω_{j_{2}j_{3}...j_{r−1}}^{σ} ∧ ω_{j_{1}i})

−\frac{r − 1}{r}pdf^{i}_{σ}^{j_{1}j_{2}...j_{r−1}} ∧ ω_{j_{2}j_{3}...j_{r−1}}^{σ} ∧ ω_{j_{1}i}.

Then

Θ = fω_{0} + (f^{i}_{σ}ω^{σ} + f^{i}_{σ}^{j_{1}}ω_{j_{1}}^{σ} + f^{i}_{σ}^{j_{1}j_{2}}ω_{j_{1}j_{2}}^{σ}

+... + f^{i}_{σ}^{j_{1}j_{2}...j_{r−2}}ω_{j_{1}j_{2}...j_{r−2}}^{σ}) ∧ ω_{i} + f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ} ∧ ω_{i}

+\frac{r − 1}{r}d_{k}(f^{k}_{σ}^{ij_{1}j_{2}...j_{r−2}} − f^{i}_{σ}^{kj_{1}j_{2}...j_{r−2}})ω_{j_{1}j_{2}...j_{r−2}}^{σ} ∧ ω_{i}

+\frac{r − 1}{r}d(f^{i}_{σ}^{j_{1}j_{2}...j_{r−1}}ω_{j_{2}j_{3}...j_{r−1}}^{σ} ∧ ω_{j_{1}i})

−\frac{r − 1}{r}pdf^{i}_{σ}^{j_{1}j_{2}...j_{r−1}} ∧ ω_{j_{2}j_{3}...j_{r−1}}^{σ} ∧ ω_{j_{1}i}.

We can apply the same decomposition to the term

(f^{i}_{σ}^{j_{1}j_{2}...j_{r−2}} + \frac{r − 1}{r}d_{k}(f^{k}_{σ}^{ij_{1}j_{2}...j_{r−2}} − f^{i}_{σ}^{kj_{1}j_{2}...j_{r−2}}) ω_{j_{1}j_{2}...j_{r−2}}^{σ} ∧ ω_{i}

defined on V^{r}, etc. After r − 1 steps we obtain a form on V^{2r−1},

\begin{matrix}Θ = fω_{0}+(f^{i}_{σ}ω^{σ}+f̃_{σ}^{ij_{1}}ω_{j_{1}}^{σ} +f̃_{σ}^{ij_{1}j_{2}}ω_{j_{1}j_{2}}^{σ} +...+f̃_{σ}^{ij_{1}j_{2}...j_{r−2}}ω_{j_{1}j_{2}...j_{r−2}}^{σ} \\ +f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ}) ∧ ω_{i} + dη + μ, \end{matrix} (14)

with symmetric coefficients f̃_{σ}^{ij_{1}}, f̃_{σ}^{ij_{1}j_{2}}, ..., f̃_{σ}^{ij_{1}j_{2}...j_{r−2}}, f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}. From formula (14) we have

p_{1}dΘ = (\frac{∂f}{∂y^{σ}} ω^{σ}+ \frac{∂f}{∂y_{k_{1}}^{σ}} ω_{k_{1}}^{σ}+ \frac{∂f}{∂y_{k_{1}k_{2}}^{σ}} ω_{k_{1}k_{2}}^{σ}+...+ \frac{∂f}{∂y_{k_{1}k_{2}...k_{r}}^{σ}} ω_{k_{1}k_{2}...k_{r}}^{σ})∧ ω_{0}

−(d_{i}f^{i}_{σ}ω^{σ}+d_{i}f̃_{σ}^{ij_{1}}ω_{j_{1}}^{σ}+d_{i}f̃_{σ}^{ij_{1}j_{2}}ω_{j_{1}j_{2}}^{σ}+...+d_{i}f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ})∧ ω_{0}

−(f^{i}_{σ}ω_{i}^{σ}+f̃_{σ}^{ij_{1}}ω_{ij_{1}}^{σ} +f̃_{σ}^{ij_{1}j_{2}}ω_{ij_{1}j_{2}}^{σ} +...+f̃_{σ}^{ij_{1}j_{2}...j_{r−2}}ω_{ij_{1}j_{2}...j_{r−2}}^{σ}

+f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}ω_{ij_{1}j_{2}...j_{r−1}}^{σ}) ∧ ω_{0}.

82 D. KRUPKA Collecting together the terms with the same elements of the basis of (n + 1)-forms, we obtain

p_{1}dΘ = (\frac{∂f}{∂y^{σ}} − d_{i}f̃_{σ}^{i}) ω^{σ} ∧ ω_{0} + (\frac{∂f}{∂y_{j_{1}}^{σ}} − d_{i}f̃_{σ}^{ij_{1}} − f̃_{σ}^{j_{1}}) ω_{j_{1}}^{σ} ∧ ω_{0}

+ (\frac{∂f}{∂y_{j_{1}j_{2}}^{σ}} − d_{i}f̃_{σ}^{ij_{1}j_{2}} − f̃_{σ}^{j_{2}j_{1}}) ω_{j_{1}j_{2}}^{σ} ∧ ω_{0}

+... + (\frac{∂f}{∂y_{j_{1}j_{2}...j_{r−1}}^{σ}} − d_{i}f̃_{σ}^{ij_{1}j_{2}...j_{r−1}} − f̃_{σ}^{j_{r−1}j_{1}j_{2}...j_{r−2}}) ω_{j_{1}j_{2}...j_{r−1}}^{σ} ∧ ω_{0}

+ (\frac{∂f}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} − f̃_{σ}^{j_{r}j_{1}j_{2}...j_{r−1}}) ω_{j_{1}j_{2}...j_{r}}^{σ} ∧ ω_{0}.

Consequently, p_{1}dΘ is ω^{σ}-generated if and only if

\frac{∂f}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} − f̃_{σ}^{j_{r}j_{1}j_{2}...j_{r−1}} = 0,

\frac{∂f}{∂y_{j_{1}j_{2}...j_{r−1}}^{σ}} − d_{i}f̃_{σ}^{ij_{1}j_{2}...j_{r−1}} − f̃_{σ}^{j_{r−1}j_{1}j_{2}...j_{r−2}} = 0,

\frac{∂f}{∂y_{j_{1}j_{2}...j_{r−2}}^{σ}} − d_{i}f̃_{σ}^{ij_{1}j_{2}...j_{r−2}} − f̃_{σ}^{j_{r−2}j_{1}j_{2}...j_{r−3}} = 0,

...

\frac{∂f}{∂y_{j_{1}j_{2}}^{σ}} − d_{i}f̃_{σ}^{ij_{1}j_{2}} − f̃_{σ}^{j_{2}j_{1}} = 0,

\frac{∂f}{∂y_{j_{1}}^{σ}} − d_{i}f̃_{σ}^{ij_{1}} − f̃_{σ}^{j_{1}} = 0.

These equations have a unique solution

f̃_{σ}^{j_{r}j_{1}j_{2}...j_{r−1}} = \frac{∂f}{∂y_{j_{1}j_{2}...j_{r}}^{σ}} ,

f̃_{σ}^{j_{r−1}j_{1}j_{2}...j_{r−2}} = \frac{∂f}{∂y_{j_{1}j_{2}...j_{r−1}}^{σ}} − d_{i_{r}} \frac{∂f}{∂y_{j_{1}j_{2}...j_{r−1}i_{r}}^{σ}} ,

f̃_{σ}^{j_{r−2}j_{1}j_{2}...j_{r−3}} = \frac{∂f}{∂y_{j_{1}j_{2}...j_{r−2}}^{σ}} − d_{i_{r−1}} \frac{∂f}{∂y_{j_{1}j_{2}...j_{r−2}i_{r−1}}^{σ}}

+d_{i_{r−1}}d_{i_{r}} \frac{∂f}{∂y_{j_{1}j_{2}...j_{r−2}i_{r−1}i_{r}}^{σ}} ,

...

f̃_{σ}^{j_{2}j_{1}} = \frac{∂f}{∂y_{j_{1}j_{2}}^{σ}} − d_{i_{3}} \frac{∂f}{∂y_{j_{1}j_{2}i_{3}}^{σ}} + d_{i_{3}}d_{i_{4}} \frac{∂f}{∂y_{j_{1}j_{2}i_{3}i_{4}}^{σ}}

−... + (−1)^{r−2}d_{i_{3}}d_{i_{4}}...d_{i_{r}} \frac{∂f}{∂y_{j_{1}j_{2}i_{3}i_{4}...i_{r}}^{σ}} ,

f̃_{σ}^{j_{1}} = \frac{∂f}{∂y_{j_{1}}^{σ}} − d_{i_{2}} \frac{∂f}{∂y_{j_{1}i_{2}}^{σ}} + d_{i_{2}}d_{i_{3}} \frac{∂f}{∂y_{j_{1}i_{2}i_{3}}^{σ}}

−... + (−1)^{r−1}d_{i_{2}}d_{i_{3}}...d_{i_{r}} \frac{∂f}{∂y_{j_{1}i_{2}i_{3}i_{4}...i_{r}}^{σ}} .

This proves Lemma 3. □

Any form Θ , satisfying properties (a) and (b) of Lemma 3, is called a Lepage equivalent of f. The form

Θ_{f} = fω_{0} + (f^{i}_{σ}ω^{σ} + f̃_{σ}^{ij_{1}}ω_{j_{1}}^{σ} + f̃_{σ}^{ij_{1}j_{2}}ω_{j_{1}j_{2}}^{σ} + ... + f̃_{σ}^{ij_{1}j_{2}...j_{r−2}}ω_{j_{1}j_{2}...j_{r−2}}^{σ}
+f̃_{σ}^{ij_{1}j_{2}...j_{r−1}}ω_{j_{1}j_{2}...j_{r−1}}^{σ}) ∧ ω_{i}

is the principal Lepage equivalent of f.

Note that Lepage equivalents Θ of f are defined by prescribing some properties of the exterior derivative dΘ. The meaning of any Θ for the total divergence equation consists in the structure of the form p_{1}dΘ. Computing p_{1}dΘ, we obtain the Euler-Lagrange form

p_{1}dΘ = (\frac{∂f}{∂y^{σ}} − d_{i}f^{i}_{σ}) ω^{σ} ∧ ω_{0} = E_{σ}(f)ω^{σ} ∧ ω_{0}.

5. The fibered homotopy operator

Let U ⊂ ℝ^{n} be an open set, let W ⊂ ℝ^{m} be an open ball with center at the origin, and let ζ : U → U × W be the zero section. We define a mapping χ : [0, 1] × U × W → U × W by

χ(s, (x^{i}, y^{σ})) = (x^{i}, sy^{σ}).

Then

χ^{∗}dx^{i} = dx^{i}, χ^{∗}dy^{σ} = y^{σ}ds + sdy^{σ}.  (15)

For any k-form ρ on U × W, where k ≥ 1, consider the pull-back χ^{∗}ρ, which is a k-form on the set [0, 1] × U × W. Obviously, there exists a unique decomposition

χ^{∗}ρ = ds ∧ ρ^{(0)}(s) + ρ′(s)  (16)

such that the (k − 1)-form ρ^{(0)}(s) and k-form ρ′(s) do not contain ds. Note that by (15), ρ′(s) arises from ρ by replacing each factor dy^{σ} by sdy^{σ}, and by replacing each coefficient f in ρ by f ∘ χ; the factors dx^{i} remain unchanged. Thus, ρ′(s) obeys

ρ′(1) = ρ, ρ′(0) = π^{∗}ζ^{∗}ρ.  (17)

Define

Iρ = ∫ ρ^{(0)}(s),  (18)

where the expression on the right means integration of the coefficients in the form ρ^{(0)}(s) over s from 0 to 1. If f : ×W → ℝ is a function, we define

If = 0.

The mapping ρ → Iρ is called the fibered homotopy operator.

We prove the following result.

Lemma 4. For any differential k-form ρ on U × W,

ρ = Idρ + dIρ + π^{∗}ζ^{∗}ρ.

Proof. 1. Let k = 0. If f is a function, we have by (15)

χ^{∗}df = (\frac{∂f}{∂x^{i}} ∘ χ) dx^{i} + (\frac{∂f}{∂y^{σ}} ∘ χ) (y^{σ}ds + sdy^{σ}),

and

Idf = y^{σ} ∫ (\frac{∂f}{∂y^{σ}} ∘ χ) ds.

Now the identity

f − π^{∗}ζ^{∗}f = f ∘ χ|_{s=1} − f ∘ χ|_{s=0} =∫ \frac{d(f ∘ χ)}{ds}ds = y^{σ}∫ (\frac{∂f}{∂y^{σ}} ∘ χ)ds

gives the result.

2. Let k = 1. ρ has an expression

ρ = A_{i}dx^{i} + B_{σ}dy^{σ}.

Thus

χ^{∗}ρ = y^{σ}(B_{σ} ∘ χ)ds + (A_{i} ∘ χ)dx^{i} + (B_{σ} ∘ χ)sdy^{σ},

and

χ^{∗}dρ = ds ∧ (−d(y^{σ}(B_{σ} ∘ χ)) + \frac{∂(A_{i} ∘ χ)}{∂s}dx^{i} + \frac{∂((B_{σ} ∘ χ)s)}{∂s}dy^{σ})

+ (\frac{∂(A_{i} ∘ χ)}{∂x^{j}}dx^{j} + \frac{∂(A_{i} ∘ χ)}{∂y^{ν}}dy^{ν}) ∧ dx^{i}

+ (\frac{∂(B_{σ} ∘ χ)}{∂x^{j}}dx^{j} + \frac{∂(B_{σ} ∘ χ)}{∂y^{ν}}dy^{ν}) ∧ dy^{σ},

THE TOTAL DIVERGENCE EQUATION 85

hence Iρ = y^{σ} ∫ B_{σ} ∘ χ · ds,

Idρ = ∫ (\frac{∂(A_{i} ∘ χ)}{∂s} − \frac{∂(y^{ν} · B_{ν} ∘ χ)}{∂x^{i}}) ds · dx^{i}

+ ∫ (\frac{∂((B_{σ} ∘ χ)s)}{∂s} − \frac{∂(y^{ν} · B_{ν} ∘ χ)}{∂y^{σ}}) ds · dy^{σ},

and

dIρ = y^{σ} ∫ \frac{∂}{∂x^{i}} (B_{σ} ∘ χ)ds · dx^{i} + ∫ \frac{∂}{∂y^{σ}} (y^{ν} · B_{ν} ∘ χ)ds · dy^{σ}.

Consequently,

Idρ + dIρ = ∫ (\frac{∂(A_{i} ∘ χ)}{∂s}) ds · dx^{i} + ∫ (\frac{∂((B_{σ} ∘ χ)s)}{∂s}) ds · dy^{σ}

= ρ − π^{∗}ζ^{∗}ρ.

3. Let k ≥ 2 . Write ρ in the form

ρ = dx^{i} ∧ Φ_{i} + dy^{σ} ∧ Ψ_{σ},

and we define differential forms Φ_{i}^{(0)}(s), Φ′_{i}(s), Ψ_{σ}^{(0)}(s), Ψ′_{σ}(s) by the following decompositions

χ^{∗}Φ_{i} = ds ∧ Φ_{i}^{(0)}(s) + Φ′_{i}(s), χ^{∗}Ψ_{σ} = ds ∧ Ψ_{σ}^{(0)}(s) + Ψ′_{σ}(s). Then we get

χ^{∗}ρ = ds ∧ (−dx^{i} ∧ Φ_{i}^{(0)}(s) − sdy^{σ} ∧ Ψ_{σ}^{(0)}(s) + y^{σ}Ψ′_{σ}(s))

+dx^{i} ∧ Φ′_{i}(s) + sdy^{σ} ∧ Ψ′_{σ}(s).

Thus,

Iρ = −dx^{i} ∧ ∫ Φ_{i}^{(0)}(s) − dy^{σ} ∧ ∫ sΨ_{σ}^{(0)}(s) + y^{σ} ∫ Ψ′_{σ}(s)ds.

To determine Idρ, we compute χ^{∗}dρ. We get

χ^{∗}dρ = dχ^{∗}ρ = ds ∧ (−dx^{i} ∧ dΦ_{i}^{(0)}(s) + dx^{i} ∧ \frac{∂Φ′_{i}(s)}{∂s}

−sdy^{σ} ∧ dΨ_{σ}^{(0)}(s) − dy^{σ} ∧ Ψ′_{σ}(s)

−y^{σ}dΨ′_{σ}(s) + dy^{σ} ∧ \frac{∂(sΨ′_{σ}(s))}{∂s})

−dx^{i} ∧ (dx^{j} ∧ \frac{∂Φ′_{i}(s)}{∂x^{j}} + dy^{ν} ∧ \frac{∂Φ′_{i}(s)}{∂y^{ν}})

−dy^{σ} ∧ (dx^{j} ∧ \frac{∂(sΨ′_{σ}(s))}{∂x^{j}} + dy^{ν} ∧ \frac{∂(sΨ′_{σ}(s))}{∂y^{ν}}) ,

(19) where ∂η(s)/∂s denotes the form, arising from η(s) by differentiation of the coefficients with respect to s. Now by (19) and (18),

Idρ = −dx^{i} ∧ ∫ dΦ_{i}^{(0)}(s) − dy^{σ} ∧ ∫ sdΨ_{σ}^{(0)}(s) − dy^{σ} ∧ ∫ Ψ′_{σ}(s)

−y^{σ} ∫ dΨ′_{σ}(s) + dx^{i} ∧ ∫ \frac{∂Φ′_{i}(s)}{∂s} + dy^{σ} ∧ ∫ \frac{∂(sΨ′_{σ}(s))}{∂s}.

It is important to notice that the exterior derivatives dΦ_{i}^{(0)}(s), dΨ_{σ}^{(0)}(s), and dΨ′_{σ}(s) have the meaning of the derivatives with respect to x^{i}, y^{σ} (the terms containing ds are canceled; see the definition of I (16), (18)).

Now we easily get

Idρ + dIρ = dx^{i} ∧ ∫ \frac{∂Φ′_{i}(s)}{∂s} + dy^{σ} ∧ ∫ \frac{∂(sΨ′_{σ}(s))}{∂s}.

Remembering that the integral symbol denotes integration of coefficients in the corresponding forms with respect to s from 0 to 1, and using (17), one obtains the final formula

Idρ + dIρ = dx^{i} ∧ Φ_{i} + dy^{σ} ∧ Ψ_{σ} − dx^{i} ∧ π^{∗}ζ^{∗}Φ_{i} = ρ − π^{∗}ζ^{∗}ρ.

□

6. The total divergence equation

We now prove two theorems, describing solutions of the total divergence equation

d_{i}g^{i} = f,  (20)

where f : V^{r} → ℝ is a given function. By a solution of this equation we mean any system of functions g = g^{i} , defined on V^{s} for some nonnegative integer s, satisfying condition (20).

Lemma 5. If the total divergence equation has a solution defined on V^{s} and s ≥ r + 1, then it has a solution defined on V^{s−1}.

Proof. Suppose that we have a solution g = g^{i} of equation (20), defined on V^{s}. Since f and g^{i} do not depend on y_{j_{1}j_{2}...j_{s}j_{s+1}}^{ν}, the functions g^{i} satisfy

\begin{matrix}\frac{∂f}{∂y_{i_{1}i_{2}...i_{s+1}}^{σ}} = d_{i} \frac{∂g^{i}}{∂y_{i_{1}i_{2}...i_{s+1}}^{σ}} + \frac{1}{s + 1} (\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{s+1}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}...i_{s+1}}^{σ}} \\ +\frac{∂g^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}...i_{s+1}}^{σ}} + ... + \frac{∂g^{i_{s+1}}}{∂y_{i_{2}i_{3}...i_{s}i_{1}}^{σ}} ) \\ = \frac{1}{s + 1} (\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{s+1}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}...i_{s+1}}^{σ}} + \frac{∂g^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}...i_{s+1}}^{σ}} \\ +... + \frac{∂g^{i_{s+1}}}{∂y_{i_{2}i_{3}...i_{s}i_{1}}^{σ}} ) = 0. \end{matrix}  (21)

Analogously, with the help of (21),

\begin{matrix}\frac{∂f}{∂y_{i_{1}i_{2}...i_{s}}^{σ}} = d′_{i} \frac{∂g^{i}}{∂y_{i_{1}i_{2}...i_{s}}^{σ}} + \frac{∂^{2}g^{i}}{∂y_{j_{1}j_{2}...j_{s}}^{ν}∂y_{i_{1}i_{2}...i_{s}}^{σ}} y_{j_{1}j_{2}...j_{s}i}^{ν} \\ +\frac{1}{s} (\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{s}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}...i_{s}}^{σ}} + \frac{∂g^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}...i_{s}}^{σ}} + ... + \frac{∂g^{i_{s}}}{∂y_{i_{2}i_{3}...i_{s−1}i_{1}}^{σ}} ) \\ = d′_{i} \frac{∂g^{i}}{∂y_{i_{1}i_{2}...i_{s}}^{σ}} + \frac{1}{s} (\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{s}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}...i_{s}}^{σ}} + \frac{∂g^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}...i_{s}}^{σ}} \\ +... + \frac{∂g^{i_{s}}}{∂y_{i_{2}i_{3}...i_{s−1}i_{1}}^{σ}} ) = 0 \end{matrix} (22)

because s ≥ r + 1. From (21) and (22) we obtain

\begin{matrix}f = \frac{∂g^{i}}{∂x^{i}} + \frac{∂g^{j_{1}}}{∂y^{ν}} y_{j_{1}}^{ν} + \frac{∂g^{j_{2}}}{∂y_{j_{1}}^{ν}} y_{j_{1}j_{2}}^{ν} + ... + \frac{∂g^{j_{s−1}}}{∂y_{j_{1}j_{2}...j_{s−2}}^{ν}} y_{j_{1}j_{2}...j_{s−2}j_{s−1}}^{ν} \\ −d′_{i} \frac{∂g^{i}}{∂y_{j_{1}j_{2}...j_{s}}^{ν}} y_{j_{1}j_{2}...j_{s}}^{ν}. \end{matrix} (23)

We already know that as a consequence of condition (1),

g^{i} = g_{0}^{i} + g_{1}^{i} + g_{2}^{i} + ... + g_{n−1}^{i},  (24)

where g_{p}^{i} is a homogeneous polynomial of degree p (proof of Lemma 2, part 3). Note that substituting from (24) to (23), we get the sum of homogeneous polynomials of degree p in y_{l_{1}l_{2}...l_{s}}^{σ},

\frac{∂g_{p}^{i}}{∂x^{i}} +\frac{∂g_{p}^{j_{1}}}{∂y^{ν}} y_{j_{1}}^{ν}+\frac{∂g_{p}^{j_{2}}}{∂y_{j_{1}}^{ν}} y_{j_{1}j_{2}}^{ν}+...+\frac{∂g_{p}^{j_{s−1}}}{∂y_{j_{1}j_{2}...j_{s−2}}^{ν}} y_{j_{1}j_{2}...j_{s−2}j_{s−1}}^{ν}−d′_{i} \frac{∂g_{p}^{i}}{∂y_{j_{1}j_{2}...j_{s}}^{ν}} y_{j_{1}j_{2}...j_{s}}^{ν}.

Then setting y_{j_{1}j_{2}...j_{s}}^{ν} = 0 on both sides, we obtain

f = \frac{∂g_{0}^{i}}{∂x^{i}} + \frac{∂g_{0}^{j_{1}}}{∂y^{σ}} y_{j_{1}}^{σ} + \frac{∂g_{0}^{j_{2}}}{∂y_{j_{1}}^{σ}} y_{j_{1}j_{2}}^{σ} + ... + \frac{∂g_{0}^{j_{s−1}}}{∂y_{j_{1}j_{2}...j_{s−2}}^{σ}} y_{j_{1}j_{2}...j_{s−2}j_{s−1}}^{σ} = d_{i}g_{0}^{i}.

Thus, under the hypothesis (b), we have constructed from the solution g^{i} of the total divergence equation, defined on V^{s}, a new solution, g_{0}^{i}, defined on V^{s−1}. □

Theorem 1. Let f : V^{r} → ℝ be a function. The following two conditions are equivalent:

(a) The total divergence equation has a solution, defined on V^{r}.

(b) The function f satisfies

E_{σ}(f) = 0.

Proof. 1. Suppose that (1) has a solution g = g^{i}. Then from (2), Section 1,

\frac{∂d_{i}g^{i}}{∂y^{σ}} = d_{i} \frac{∂g^{i}}{∂y^{σ}} ,

and for every k = 1, 2, ..., r,

\frac{∂d_{i}g^{i}}{∂y_{i_{1}i_{2}...i_{k}}^{σ}} = d_{i} \frac{∂g^{i}}{∂y_{i_{1}i_{2}...i_{k}}^{σ}} + \frac{1}{k} (\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{k}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}...i_{k}}^{σ}} + \frac{∂g^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}...i_{k}}^{σ}}

+... + \frac{∂g^{i_{k}}}{∂y_{i_{2}i_{3}...i_{k−1}i_{1}}^{σ}} ) .

Using these formulas, we can compute the expression E_{σ}(f) = E_{σ}(d_{i}g^{i}) in several steps. First, we have

E_{σ}(d_{i}g^{i}) = d_{i_{1}} (\frac{∂g^{i_{1}}}{∂y^{σ}} − \frac{∂d_{s}g^{s}}{∂y_{i_{1}}^{σ}} + d_{i_{2}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}}^{σ}}

−... + (−1)^{r}d_{i_{2}}d_{i_{3}}...d_{i_{r}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} )

= d_{i_{1}}d_{i_{2}} (− \frac{∂g^{i_{2}}}{∂y_{i_{1}}^{σ}} + \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}}^{σ}} − d_{i_{3}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}i_{3}}^{σ}}

+... + (−1)^{r−1}d_{i_{3}}d_{i_{4}}...d_{i_{r−1}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r−1}}^{σ}} + (−1)^{r}d_{i_{3}}d_{i_{4}}...d_{i_{r}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} ) .

Second, using symmetrization,

E_{σ}(d_{i}g^{i}) = d_{i_{1}}d_{i_{2}} (− \frac{∂g^{i_{2}}}{∂y_{i_{1}}^{σ}} + d_{i} \frac{∂g^{i}}{∂y_{i_{1}i_{2}}^{σ}} + \frac{1}{2} (\frac{∂g^{i_{1}}}{∂y_{i_{2}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}}^{σ}} ) − d_{i_{3}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}i_{3}}^{σ}}

+... + (−1)^{r−1}d_{i_{3}}d_{i_{4}}...d_{i_{r−1}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r−1}}^{σ}} + (−1)^{r}d_{i_{3}}d_{i_{4}}...d_{i_{r}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} )

= d_{i_{1}}d_{i_{2}}d_{i_{3}} (\frac{∂g^{i_{3}}}{∂y_{i_{1}i_{2}}^{σ}} − \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}i_{3}}^{σ}} + ... + (−1)^{r−1}d_{i_{4}}d_{i_{5}}...d_{i_{r−1}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r−1}}^{σ}}

+(−1)^{r}d_{i_{4}}d_{i_{5}}...d_{i_{r}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} ) .

Third, again with the help of symmetrization,

E_{σ}(d_{i}g^{i}) = d_{i_{1}}d_{i_{2}}d_{i_{3}} (− d_{i_{4}} \frac{∂g^{i_{4}}}{∂y_{i_{1}i_{2}i_{3}}^{σ}} + d_{i_{4}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}i_{3}i_{4}}^{σ}}

−... + (−1)^{r−1}d_{i_{4}}d_{i_{5}}...d_{i_{r−1}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r−1}}^{σ}} + (−1)^{r}d_{i_{4}}d_{i_{5}}...d_{i_{r}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} )

= d_{i_{1}}d_{i_{2}}d_{i_{3}}d_{i_{4}} (− \frac{∂g^{i_{4}}}{∂y_{i_{1}i_{2}i_{3}}^{σ}} + \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}i_{3}i_{4}}^{σ}}

−... + (−1)^{r−1}d_{i_{5}}d_{i_{6}}...d_{i_{r−1}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r−1}}^{σ}} + (−1)^{r}d_{i_{5}}d_{i_{6}}...d_{i_{r}} \frac{∂d_{s}g^{s}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} ).

We continue this process, and obtain after r − 1 steps

E_{σ}(d_{i}g^{i}) = (−1)^{r}d_{i_{1}}d_{i_{2}}...d_{i_{r−1}}d_{i_{r}}d_{i} \frac{∂g^{i}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} .  (25)

But since f is defined on V^{r}, the solution g of equation (1) necessarily satisfies

\frac{∂g^{k}}{∂y_{i_{1}i_{2}...i_{r}}^{σ}} + \frac{∂g^{i_{1}}}{∂y_{ki_{2}i_{3}...i_{r}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}ki_{3}i_{4}...i_{r}}^{σ}} + ... + \frac{∂g^{i_{r−1}}}{∂y_{i_{1}i_{2}...i_{r−2}ki_{r}}^{σ}} + \frac{∂g^{i_{r}}}{∂y_{i_{1}i_{2}...i_{r−1}k}^{σ}} = 0.

Using this formula in (25) we see that condition (b) is satisfied.

2. From now on, we suppose that condition (b) is satisfied. We want to show that there exist functions g^{i} : V^{r} → ℝ such that d_{i}g^{i} = f, or, in an explicit form,

\frac{∂g^{i}}{∂x^{i}} + \frac{∂g^{j_{1}}}{∂y^{σ}} y_{j_{1}}^{σ} + \frac{∂g^{j_{2}}}{∂y_{j_{1}}^{σ}} y_{j_{1}j_{2}}^{σ} + ... + \frac{∂g^{j_{r}}}{∂y_{j_{1}j_{2}...j_{r−1}}^{σ}} y_{j_{1}j_{2}...j_{r−1}j_{r}}^{σ} = f,

and

\frac{∂g^{i_{1}}}{∂y_{i_{2}i_{3}...i_{r+1}}^{σ}} + \frac{∂g^{i_{2}}}{∂y_{i_{1}i_{3}i_{4}...i_{r+1}}^{σ}} + \frac{∂g^{i_{3}}}{∂y_{i_{2}i_{1}i_{4}i_{5}...i_{r+1}}^{σ}}

+... + \frac{∂g^{i_{r}}}{∂y_{i_{2}i_{3}...i_{r−1}i_{1}i_{r+1}}^{σ}} + \frac{∂g^{i_{r+1}}}{∂y_{i_{2}i_{3}...i_{r−1}i_{r}i_{1}}^{σ}} = 0.

Let I be the fibered homotopy operator for differential form on V^{2r}, associated with the projection π^{2r} : V^{2r} → U (Section 5). We have

Θ_{f} = IdΘ_{f} + dIΘ_{f} + Θ_{0} = Ip_{1}dΘ_{f} + Ip_{2}dΘ_{f} + dIΘ_{f} + Θ_{0},

where Θ_{0} is an n-form, projectable on U. In this formula, p_{1}dΘ_{f} = 0 by hypothesis, and we may suppose that Θ_{0} = dϑ_{0} (on U). Moreover hΘ_{f} = hd(IΘ_{f} + ϑ_{0}) = fω_{0}. Defining functions g^{i} on V^{s}, where s ≤ 2r, by the condition

h(IΘ_{f} + ϑ_{0}) = g^{i}ω_{i},

we see we have constructed a solution of the total divergence equation d_{i}g^{i} = f. Explicitly,

\frac{∂g^{i}}{∂x^{i}} + \frac{∂g^{j_{1}}}{∂y^{σ}} y_{j_{1}}^{σ} + \frac{∂g^{j_{2}}}{∂y_{j_{1}}^{σ}} y_{j_{1}j_{2}}^{σ} + ... + \frac{∂g^{j_{s+1}}}{∂y_{j_{1}j_{2}...j_{s}}^{σ}} y_{j_{1}j_{2}...j_{s}j_{s+1}}^{σ} = f.  (26)

Note, however, that in general, we have not yet proved that the total divergence equation has a solution defined on V^{r}.

If s ≤ r, formula (26) shows that condition (a) holds. If s ≥ r + 1, we apply Lemma 5 several times, and obtain a solution of equation (26) defined on V^{r}.

This concludes the proof. □

Combining Theorem 1 and Lemma 2, we can easily describe all solutions of the total divergence equations d_{i}g^{i} = f such that the right side f satisfies the integrability condition E_{σ}(f) = 0. In particular, we show that a convenient description of solutions arises when we interpret them as some differential forms.

Theorem 2. Let f : V^{r} → ℝ be a function such that E_{σ}(f) = 0. The following conditions are equivalent:

(a) g = g^{i} is a solution of the total divergence equation d_{i}g^{i} = f, defined on V^{r}.

(b) The form η = g^{i}ω_{i} has a projectable extension.

(c) g^{i} is given by

g^{i} = \frac{1}{(n − 1)!}ε^{ii_{1}i_{2}...i_{n−1}} (A_{i_{1}i_{2}...i_{n−1}}

+ ∑_{k=1}^{n−1} \frac{(n − 1)!}{k!(n − 1 − k)!}A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{k}i_{k+1}i_{k+2}...i_{n−1}}^{J_{k}}y_{J_{1}i_{1}}^{σ_{1}}y_{J_{2}i_{2}}^{σ_{2}}...y_{J_{k}i_{k}}^{σ_{k}}) ,

where A_{i_{1}i_{2}...i_{n−1}}, A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{k}i_{k+1}i_{k+2}...i_{n−1}}^{J_{k}} are arbitrary functions on V^{r−1}, antisymmetric in all indices and multi-indices, and |J_{1}|, |J_{2}|, ..., |J_{n−1}| = r − 1.

Proof. Assertions (a) and (b) are equivalent by Lemma 2. To show that (c) is equivalent with (b), we consider any (n − 1)-form on V^{r−1},

μ = \frac{1}{(n − 1)!}A_{i_{1}i_{2}...i_{n−1}}dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{n−1}}

+ ∑_{k=1}^{n−2} \frac{1}{k!(n − 1 − k)!}A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{k}i_{k+1}i_{k+2}...i_{n−1}}^{J_{k}}dy_{J_{1}}^{σ_{1}} ∧ dy_{J_{2}}^{σ_{2}} ∧ ... ∧ dy_{J_{k}}^{σ_{k}}

∧dx^{i_{k+1}} ∧ dx^{i_{k+2}} ∧ ... ∧ dx^{i_{n−1}}

+\frac{1}{(n − 1)!}A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{n−1}}^{J_{n−1}}dy_{J_{1}}^{σ_{1}} ∧ dy_{J_{2}}^{σ_{2}} ∧ ... ∧ dy_{J_{n−1}}^{σ_{n−1}} + τ,

where τ is a contact form. μ has the horizontal component

hμ = \frac{1}{(n − 1)!} (A_{i_{1}i_{2}...i_{n−1}}

+ ∑_{k=1}^{n−1} \frac{(n − 1)!}{k!(n − 1 − k)!}A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{k}i_{k+1}i_{k+2}...i_{n−1}}^{J_{k}}y_{J_{1}i_{1}}^{σ_{1}}y_{J_{2}i_{2}}^{σ_{2}}...y_{J_{k}i_{k}}^{σ_{k}})

·dx^{i_{1}} ∧ dx^{i_{2}} ∧ ... ∧ dx^{i_{k}} ∧ dx^{i_{k+1}} ∧ ... ∧ dx^{i_{n−1}}.

Transforming hμ we obtain hμ = g^{i}ω_{i}, where

g^{i} = \frac{1}{(n − 1)!}ε^{ii_{1}i_{2}...i_{n−1}} (A_{i_{1}i_{2}...i_{n−1}}

+ ∑_{k=1}^{n−1} \frac{(n − 1)!}{k!(n − 1 − k)!}A_{σ_{1}}^{J_{1}} _{σ_{2}}^{J_{2}}..._{σ_{k}i_{k+1}i_{k+2}...i_{n−1}}^{J_{k}}y_{J_{1}i_{1}}^{σ_{1}}y_{J_{2}i_{2}}^{σ_{2}}...y_{J_{k}i_{k}}^{σ_{k}}).

□

Note that Theorem 2 together with Lemma 2 show that solutions of the total divergence equation can be interpreted as certain differential 92 D. KRUPKA

(n − 1)-forms on V^{r−1}; the correspondence between these two objects is given by formula

η = g^{i}ω_{i} = \frac{1}{(n − 1)!}h_{j_{2}j_{3}...j_{n}}dx^{j_{2}} ∧ dx^{j_{3}} ∧ ... ∧ dx^{j_{n}},

and Lemma 1.

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THE TOTAL DIVERGENCE EQUATION 93

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Department of Algebra and Geometry, Palacky University, Olomouc, Czech Republic; Department of Mathematics, La Trobe University, Melbourne, Victoria 3086, Australia E-mail address: krupka@inf.upol.cz

Received June 27, 2006