Helmholtz Decomposition and Rotation Potentials in N-Dimensional Cartesian Coordinates

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Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

Erhard Glötzl1 , Oliver Richters2,3

1: Institute of Physical Chemistry, Johannes Kepler University Linz, Austria. 2: ZOE. Institute for Future-Fit Economies, Bonn, Germany. 3: Department of Business Administration, Economics and Law, Carl von Ossietzky University, Oldenburg, Germany.

Version 3 – July 2021

Abstract: This paper introduces a simplified method to extend the Helmholtz Decompo- sition to n-dimensional sufficiently smooth and fast decaying vector fields. The rotation is described by a superposition of n(n − 1)/2 rotations within the coordinate planes. The source potential and the rotation potential are obtained by convolving the source and rotation densities with the fundamental solutions of the Laplace equation. The rotation-free gradient of the source potential and the divergence-free rotation of the rotation potential sum to the original vector field. The approach relies on partial derivatives and a Newton potential operator and allows for a simple application of this standard method to high- dimensional vector fields, without using concepts from differential geometry and tensor calculus.

Keywords: Helmholtz Decomposition, Fundamental Theorem of Calculus, Curl Operator

Licence: Creative-Commons CC-BY-NC-ND 4.0.

1. Introduction

The Helmholtz Decomposition splits a sufficiently smooth and fast decaying vector field into an irrotational (curl-free) and a solenoidal (divergence-free) vector field. In R3, this ‘Fundamental Theorem of Vector Calculus’ allows to calculate a scalar and a vector potential that serve as antiderivatives of the gradient and the curl operator. This tool is indispensable for many problems in mathematical physics (Dassios and Lindell, 2002; Kustepeli, 2016; Sprössig, 2009; Suda, 2020), arXiv:2012.13157v3 [math-ph] 16 Jul 2021 but has also found applications in animation, computer vision, robotics (Bhatia et al., 2013), or for describing a ‘quasi-potential’ landscapes and Lyapunov functions for high-dimensional non-gradient systems (Suda, 2019; Zhou et al., 2012). The literature review in Section 2 summarizes the classical Helmholtz Decomposition in R3 and the Helmholtz–Hodge Decomposition that generalizes the operator curl to higher dimensions using concepts of differential geometry and tensor calculus. Section 3 introduces a much simpler generalization to higher dimensions in Cartesian coordinates and states the Helmholtz Decomposition Theorem using novel differential operators. This approach relies on partial derivatives and convolution integrals and avoids the need for concepts from differential geometry and tensor calculus. Section 4 defines these operators to derive a source density and a n rotational density describing the 2 basic rotations within the 2-dimensional coordinate planes. By n solving convolution integrals, a scalar source potential and 2 rotation potentials can be obtained, 2 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates and the original vector can be decomposed into a rotation-free vector and a divergence-free vector by superposing the gradient of the source potential with the rotation of the rotation potentials. Section 5 states some propositions and proves the theorem. Section 6 concludes.

2. Literature review

2.1. Classical Helmholtz Decomposition in R3 In its classical formulation,1 the Helmholtz Decomposition decomposes a vector field f ∈ C2(R3, R3) that decays faster than |x|−2 for |x| → ∞ into an irrotational (curl-free) vector field g(x) = − grad Φ(x) with a scalar potential Φ ∈ C3(R3, R) and a solenoidal (divergence-free) vector field r(x) = curl A(x) with a vector potential A ∈ C3(R3, R3) such that f (x) = g(x) + r(x). The potentials can be derived by calculating the source density γ(x) and the rotation density ρ(x):2 γ(x) = div f (x) = div g(x), ρ(x) = curl f (x) = curl r(x). (1) The convolution with the fundamental solutions of the Laplace equation provides the potentials: 1 γ(ξ) 1 ρ(ξ) Φ(x) = dξ3, A(x) = dξ3. (2) 4π R3 |x − ξ| 4π R3 |x − ξ| The Helmholtz decomposition$ of f is given as: $ f (x) = g(x) + r(x) with g(x) = − grad Φ(x) and r(x) = curl A(x). (3)

2.2. Previous extensions to higher dimensions For generalizing the Helmholtz Decomposition to higher-dimensional manifolds, divergence and gradient can straightforwardly be extended to any dimension n, but not the operator curl and the cross product. This lead to the Hodge Decomposition within the framework of differential forms, defining the operator curl as the Hodge dual of the anti-symmetrized gradient (Hauser, 1970; McDavid and McMullen, 2006; Tran-Cong, 1993; Vargas, 2014; Wolfram Research, 2012). In two Cartesian dimensions, curl acting on a scalar field R is a two-dimensional vector field given by " # " # ∂R ∂R ∂R ∂R curl R(x) = − , = δ2k − δ1k ; 1 ≤ k ≤ 2 . (4) ∂x2 ∂x1 ∂x1 ∂x2 Square brackets indicate vectors in Rn. The second notation will help to compare this rotation in the x1-x2-plane with the rotation in the xi-x j-plane in Eq. (25). The rotation operator acting on a two-dimensional vector field f yields a scalar field given by:

∂ f ∂ f ∂ f j ∂ f curl f (x) = 2 − 1 = − i with i = 1, j = 2. (5) ∂x1 ∂x2 ∂xi ∂x j

Here, we use the overline to indicate that curl is operating on vector fields, and curl without overline operates on scalar fields. For n = 3, the rotation of a vector field f is usually written as a pseudovector: " # ∂ f ∂ f ∂ f ∂ f ∂ f ∂ f curl f (x) = 3 − 2 , 1 − 3 , 2 − 1 . (6) ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2

1 For a historical overview of the contributions by Stokes (1849) and Helmholtz (1858), see Kustepeli (2016). 2 Note that if one is not interested in the rotation potentials, r(x) can simply be obtained after determining G(x) and g(x) by calculating r(x) = f (x) − g(x), which was the approach by Stokes (1849). 3 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

The third component (and analogously first and second) is often understood as the rotation around the x3-coordinate. In order to facilitate the extension to higher dimensions, it should better be discussed as rotation within the x1-x2-coordinate plane. Then, curl should be understood as an antisymmetric second rank tensor (Gonano and Zich, 2014; McDavid and McMullen, 2006). There n exist 2 rotations within the coordinate planes. Only for n = 3 can each of these rotations be n described as a rotation around a vector, as 2 = n if and only if n = 3. For a tensor field T with dimension n > 3 and rank k, curl T is a tensor field with dimension n and rank n − k − 1. For a scalar (rank k = 0), it consists of nn−1 entries. For a vector field in Rn (rank k = 1), curl consists of nn−2 components. Each component needs n − 2 indices and is given in Cartesian coordinates by: ! X ∂ f ∂ f X ∂ f 1 · l − m − · l curl f = 2 e1,...,en−2,l,m = e1,...,en−2,l,m , (7) e1,...,en−2 ∂x ∂x ∂x 1≤l,m≤n m l 1≤l,m≤n m

with the completely antisymmetric Levi–Civita tensor ν1...νp that is +1 (resp. −1) if the integers ν1 . . . νn are distinct and an even (resp. odd) permutation of 1 ... n, and otherwise 0. As an example, for R5, the term ∂ f5 − ∂ f4 can be found with negative sign at positions , ∂x4 ∂x5 123 231 and 312 and with positive sign at positions 132, 213 and 321. The tensor contains n(n − 1)/2 diﬀerent elements apart from sign changes, each repeated (n − 2)! times, while the rest of the nn−2 elements is zero. This enormous complexity makes higher-dimensional Helmholtz analysis challenging.

3. An alternative n-dimensional Helmholtz Decomposition Theorem

In the following, we present a simpler approach to Helmholtz Decomposition of a twice continuously differentiable vector field f (x) that decays faster than |x|−c for |x| → ∞ and c > 0. The basic idea n is to understand the rotation in dimension n as a combination of 2 rotations within the planes spanned by two of the Cartesian coordinates. In each of these planes of rotation, applying the curl n in two dimensions is sufficient. These 2 rotation densities form the upper triangle of a matrix. For tractability, we complete this to a n × n matrix by making it antisymmetric. It thus contains redundantly both the rotation in the xi-x j-plane and in the x j-xi-plane. Nevertheless, it has only n2 entries, instead of nn−2 entries for curl, and contains only the mutually different entries of this operator. We proceed along the steps shown in Figure 1. Starting from f , we calculate the scalar 2 source density γ and n basic rotation densities ρi j using the new differential operator D, consisting of the well-known divergence div and the new operator ROT (Sec. 4.1). Each basic rotation density 2 ρi j corresponds to the rotation within the xi-x j-plane. The basic rotation densities form the n - dimensional, antisymmetric rotation density ρ = ρi j = ROT f and together with γ = div f the n o n o NR density φ = γ, ρ(x) = γ, ρi j . The Newton potential operator convolves these densities with R the fundamental solutions of the Laplace equation, yielding the scalar ‘source potential’ G = N γ 2 NR 2 and n ‘basic rotation potentials’ Ri j = ρi j (Sec. 4.2). Similar to the densities, these n scalar fields Ri j can be jointly written as antisymmetric ‘rotation potential’ R = Ri j . With a new differential operator D, combining the well-known gradient grad with the new operator ROT, operating on the potential F = G, R , a rotation-free ‘gradient field’ g = grad G and a source-free ‘rotation field’ r = ROT R can be calculated (Sec. 4.3). In sum , they yield the original vector field f (x). 4 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

Figure 1: Relation between source density φ, vector field f and Newton potential F in Rn. To get the Helmholtz Decomposition of the vector field f , derive the densities applying the operator D described in Sec. 4.1. These densities are convolved with the fundamental solutions of the Laplace equation to derive scalar and rotation potentials (using the ‘Newton potential R operator N ’) as explained in Sec. 4.2. The gradient field g and the rotation field r that sum to the original field f are derived using the operator D defined in Sec. 4.3.

4. Deﬁnitions

The notation is that square brackets [ fk] indicate a vector, double square brackets ~Ri j a n × n n o 2 matrix and curly brackets γ, ~ρi j an object of dimension 1 + n . For comparison, the table in the Appendix A summarizes the definitions and important properties of the operators and variables, and shows their similarities to the well-known decomposition in dimension 3. Let f ∈ C2(Rn, Rn) be a twice continuously differentiable vector field that decays faster than |x|−c for |x| → ∞ and c > 0. f (x) = fk(x); 1 ≤ k ≤ n = f1(x),..., fn(x) . (8)

Definition 1 (Helmholtz Decomposition, gradient field and rotation field). For a vector field f ∈ C2(Rn, Rn), a ‘gradient field’ g ∈ C2(Rn, Rn) and a ‘rotation field’ r ∈ C2(Rn, Rn) are called a ‘Helmholtz Decomposition of f ’, if they sum to f , if g is gradient of some ‘gradient potential’ G, and if r is divergence-free:

f (x) = g(x) + r(x), (9) g(x) = grad G(x), (10) 0 = div r(x). (11) 5 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

4.1. From vector ﬁeld to densities To get from the vector ﬁeld f to the source and rotation densities, we start with the Jacobian Matrix J of f and its decomposition into a symmetric part S and an antisymmetric part A:

> > ∂ fi J + J J − J J = Ji j = = S + A with S = and A = . (12) ∂x j 2 2 Definition 2 (scalar source density γ). Analogously to R3, we define the ‘scalar source density’ γ ∈ C1(Rn, R) for the vector field f as trace of the Jacobian of f :

n X ∂ fi γ(x) B Tr J = = div f (x). (13) ∂x i=1 i

Definition 3 (basic rotation density operator ROTi j). We define the ‘basic rotation density operator’ 1 n n 0 n ROTi j : C (R , R ) → C (R , R) for 1 ≤ i, j ≤ n of the vector field f as 2 times the i-j-element of the antisymmetric part A of the Jacobian J:

∂ fi ∂ f j ROTi j f (x) = 2Ai j = − . (14) ∂x j ∂xi

This generalizes the two-dimensional curl in the x -x -plane of Eq. (5) given by curl f = ∂ f2 − ∂ f1 1 2 ∂x1 ∂x2 to the rotation in the xi-x j-plane, just with a different sign convention. To avoid confusion with curl, we use ROT for rotation of vector fields – and later define ROTi j analogously to curl without the overline operating on scalar fields.

Deﬁnition 4 (rotation density operator ROT). The ‘rotation density operator’ ROT: C1(Rn, Rn) → 2 C0(Rn, Rn ) is deﬁned as an antisymmetric operator containing all the basic rotation density operators: ! ∂ fi ∂ f j ROT f (x) B ROTi j f (x) = − ; 1 ≤ i, j ≤ n . (15) ∂x j ∂xi

2 Definition 5 (basic rotational densities ρi j).We define the n ‘basic rotational densities’ ρi j ∈ 2 C1(Rn, Rn) for 1 ≤ i, j ≤ n and the matrix ρ ∈ C1(Rn, Rn ) containing these n2 densities as the rotation density operators applied to the vector field f :

ρi j(x) B ROTi j f (x), (16)

ρ(x) B ρi j(x) = ρi j(x); 1 ≤ i, j ≤ n = ROTi j f (x); 1 ≤ i, j ≤ n = ROT f (x). (17)

n As ROTi j and ρi j are antisymmetric, they in fact only contain 2 independent elements, one for n each of the 2 coordinate planes.

Definition 6 (density derivative D, density φ). We define the ‘density derivative’ D: C1(Rn, Rn) → 2 C0(Rn, R1+n ) of the vector field f that combines div and ROT into one operator, and define the 2 ‘density’ φ ∈ C1(Rn, R1+n ) that combines the source and rotation densities: n o n o n o φ(x) B γ(x), ρ(x) = γ(x), ρi j(x) = D f (x) B div f (x), ROT f (x) . (18)

6 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

4.2. From densities to potentials The next step derives the potentials starting from the densities using the Newton potential operator. R R Deﬁnition 7 (Newton potential operator N ). We deﬁne the ‘Newton potential operator’ N : C0(Rn, R) → 2 n −c C (R , R) as convolution of any density Q ∈ {φ, γ, ρ, ρi j} that decays faster than |x| for |x| → ∞ and c > 0 with the fundamental solutions of the Laplace equation ∆K(x) = 0: Z Z N Q(x) = K(x, ξ) Q(ξ) dξn (19) Rn

n 2 n with the kernel K(x, ξ), using Vn = π /Γ 2 + 1 the volume of a unit n-ball and Γ(x) the gamma function,   1 log |x − ξ| − log |ξ| n = 2, K(x, ξ) =  2π  1 |x − ξ|2−n − |ξ|2−n otherwise, n(2−n)Vn We call a potential derived this way ‘Newton potential’. The decay of Q at inﬁnity guarantees the existence of the convolution integrals, similar to the 3-dimensional case, but the use of the more complex kernel compared to Eq. (2) allows to relax the restriction to ﬁelds decaying faster than |x|−c with c > 0, instead of c = 2 (Blumenthal, 1905; Petrascheck, 2015).3

Deﬁnition 8 (source potential G). We deﬁne the one-dimensional ‘source potential’ G ∈ C3(Rn, R) as application of the Newton potential operator to the source density: Z Z G(x) B N γ(x) B K(x, ξ) γ(ξ) dξn, (20) Rn

Note that compared to the case in R3, we use a diﬀerent sign convention: G(x) = −Φ(x).

2 3 n Deﬁnition 9 (basic rotation potentials Ri j). We deﬁne n ‘basic rotation potentials’ Ri j ∈ C (R , R) for 1 ≤ i, j ≤ n, corresponding to the coordinate plane spanned by xi and x j, as application of the Newton potential operator to the basic rotation densities: Z Z N n Ri j(x) B ρi j(x) = K(x, ξ) ρi j(ξ) dξ . (21) Rn

2 Deﬁnition 10 (rotation potential R). We deﬁne the ‘rotation potential’ R ∈ C3(Rn, Rn ) as matrix 2 containing the n basic rotation potentials Ri j: Z Z N N R(x) B Ri j(x) = Ri j(x); 1 ≤ i, j ≤ n = ρi j(x); 1 ≤ i, j ≤ n = ρ(x). (22)

R Here, the Newton potential operator N is applied in each component. The rotation potential R(x) is antisymmetric, thus Ri j(x) = −R ji(x) and Rii(x) = 0, and the rotation potentials therefore contains n 2 distinct components.

3 Tran-Cong (1993) shows that f (x) needs to be bounded at infinity only by O(|x|l) with l > 0 a constant using a more complicated convolution integral. Glötzl and Richters (2021) provide analytical solutions for gradient and rotation potentials and fields of several unbounded vector fields whose components are sums and products of polynomials, exponential and periodic functions. 7 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

2 Deﬁnition 11 (potential F). The ‘potential’ F ∈ C3(Rn, R1+n ) combines source and rotation potentials into one object of dimension 1 + n2: n o Z Z F(x) B G(x), R(x) = N φ(x) = N D f (x). (23)

R We know from the theory of the Poisson equation (Gilbarg and Trudinger, 1977) that ∆N Q(x) = Q(x) for Q ∈ C(Rn, Rm), which implies:

∆G(x) = γ(x), ∆Ri j(x) = ρi j(x) ∀ i, j, ∆R(x) = ρ(x), ∆F(x) = φ(x). (24)

4.3. From potentials to vector ﬁelds

1 n Definition 12 (basic rotation operator ROTi j). We define the ‘basic rotation operator’ ROTi j : C (R , R) → 0 n n C (R , R ) operating on the basic rotation potential Ri j with 1 ≤ i, j ≤ n as " # ∂Ri j ∂Ri j ROTi j Ri j(x) B 0,..., 0, + , 0,..., 0, − , 0,..., 0 (25) ∂x j ∂xi " # ∂Ri j ∂Ri j = δik − δ jk ; 1 ≤ k ≤ n , ∂x j ∂xi with the Kronecker delta δik = 1 if i = k and 0 otherwise. This operator is a generalization of curl operating on a scalar field in the two-dimensional case h i of rotations within the x -x -plane in Eq. (4) given by δ ∂R − δ ∂R ; 1 ≤ k ≤ 2 , again with a 1 2 2k ∂x1 1k ∂x2 different sign convention. It operates in the xi-x j-plane, therefore the non-zero terms in this n- dimensional vector are located at positions i and j, instead of 1 and 2 in the 2-dimensional case. Note that ROTi j Ri j(x) = ROT ji R ji(x), yielding a symmetric object.

2 Deﬁnition 13 (rotation operator ROT). We deﬁne the ‘rotation operator’ ROT: C1(Rn, Rn ) → 0 n n n C (R , R ) for each rotation potential R(x) = Ri j(x) as superposition of the 2 basic rotations in 2 n the coordinate planes. As the antisymmetric potential R(x) of dimension n contains each of the 2 basic rotations twice, we have to devide the result by 2.

n  n  1 X X ∂Rkm  ROT R(x) B ROTi j Ri j(x) =  ; 1 ≤ k ≤ n . (26) 2  ∂x  i, j=1 m=1 m

Note that this is identical to div Rkm for an antisymmetric second-rank tensor. Definition 14 (gradient field g and rotation field r). The ‘gradient field’ g ∈ C2(Rn, Rn) and the ‘rotation field’ r ∈ C2(Rn, Rn) are defined as:

g(x) = grad G(x), r(x) = ROT R(x). (27)

2 Deﬁnition 15 (derivative of the potential D). We deﬁne the ‘derivative of the potential’ D: C1(Rn, R1+n ) → C0(Rn, Rn) operating on a potential F(x) = {G(x), R(x)} as the sum of the gradient operating on the source potential G(x) and the rotation operator ROT operating on the rotation potential R(x): n o DF(x) = D G(x), R(x) B grad G(x) + ROT R(x) = g(x) + r(x). (28) 8 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

We call a potential F = {G, R} ‘antiderivative’ of f if f (x) = DF(x). Given the conditions on f (x), the potential is uniquely4 determined by Eq. (20). We will prove that Eqs. (27) and (28) are a Helmholtz Decomposition according to Deﬁnition 1.

5. Helmholtz Decomposition Theorem

Theorem 1 (Helmholtz Decomposition Theorem). Any twice continuously differentiable vector field f ∈ C2(Rn, Rn) that decays faster than |x|−c for |x| → ∞ and c > 0 can be decomposed into two vector fields, one rotation-free and one divergence-free. With the definitions of the operators in Section 4, let R G ∈ C3(Rn, R), with G(x) B N div f (x), (29) g ∈ C2(Rn, Rn), with g(x) B grad G(x), (30) 2 R R ∈ C3(Rn, Rn ), with R(x) B N ROT f (x), (31) r ∈ C2(Rn, Rn), with r(x) B ROT R(x), (32) then

g, the gradient of G, is rotation-free: 0 = ROTg(x) = ROT grad G(x), (33) r, the rotation of R, is divergence-free: 0 = div r(x) = div ROT R(x), (34) the potential {G, R} is an antiderivative of f : f (x) = DG(x), R(x) , (35) the Helmholtz Decomposition of f is given by: f (x) = g(x) + r(x). (36)

Proof. Propositions 2 and 3 show that g(x) is curl-free and r(x) is divergence-free (Eqs. 33–34). We introduce and prove Lemma 4, Corollary 5 and three operator identities as Propositions 6–8. In Proposition 9, we prove the conditions in Eqs. (35–36) that g(x) + r(x) = DF(x) equals the original vector ﬁeld f (x).

Proposition 2 (Eq. (33) of the Helmholtz Decomposition Theorem: g(x) = grad G(x) is rotation-free). For any twice continuously differentiable vector field f ∈ C2(Rn, Rn) that decays faster R than |x|−c for |x| → ∞ and c > 0, the source potential G(x) = N div f (x) as defined by Eq. (20) and its gradient g(x) = grad G(x) satisfy the following identities:

ROTg(x) = ROT grad G(x) = 0, (37) curlg(x) = curl grad G(x) = 0. (38)

Proof. Using the Deﬁnition 4 of ROT in Eq. (15) and of the components of curl in Eq. (7), and

4 Note that by Liouville’s theorem, if H is a harmonic function defined on all of Rn which is bounded above or bounded below, then H is constant, and therefore identical zero if it vanishes at infinity (Medková, 2018, p. 108). Therefore, we do not need to care about integration constants and adding harmonic functions that solve the Laplace Equation ∆H(x) = 0. If the fields do not decay sufficiently fast, alternative methods to derive Newton Potentials can be found in R the literature, see Sec. 2.1. They require a careful attention to boundary conditions because G(x) = N ∆(G(x) + H(x)) with any harmonic function H(x). 9 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates permutability of second derivatives: " # ∂G ∂G ROTg(x) = ROT grad G(x) = ROT ,..., ∂x ∂xn 1 (39) ∂ ∂G ∂ ∂G = − ; 1 ≤ i, j ≤ n = 0, ∂xi ∂x j ∂x j ∂xi ! X ∂g ∂g j 1 · i − curlg(x) = 2 e1,...,en−2,i, j e1,...,en−2 ∂x ∂x 1≤i, j≤n j i ! (40) X ∂ ∂G ∂ ∂G = 1 · − = 0. 2 e1,...,en−2,i, j ∂x ∂x ∂x ∂x 1≤i, j≤n j i i j Proposition 3 (Eq. (34) of the Helmholtz Decomposition Theorem: r(x) = ROT R(x) is divergence-free). For any twice continuously differentiable vector field f ∈ C2(Rn, Rn) that decays faster R than |x|−c for |x| → ∞ and c > 0, the rotation potential R(x) = N ROT f (x) as defined by Eqs. (21) and (22) and its rotation r(x) = ROT R(x) satisfy the following identity: div r(x) = div ROT R(x) = 0 (41)

Proof. Using definition of ROT in Eq. (26) and ROTi j in Eq. (25), exchangeability of derivatives and sums, and permutability of second derivatives: " # ∂Ri j ∂Ri j div ROTi j Ri j(x) = div δik − δ jk ; 1 ≤ k ≤ n ∂x j ∂xi n ! n  2 2  (42) X ∂ ∂Ri j ∂Ri j X  ∂ Ri j ∂ Ri j  = δik − δ jk =  −  = 0. ∂x ∂x ∂x ∂x ∂x ∂x ∂x  k=1 k j i k=1 i j j i n X 1 div r(x) = div ROT R(x) = div 2 ROTi j Ri j(x) i, j n n (43) 1 X X = 2 div ROTi j Ri j(x) = 0 = 0. i, j i, j Lemma 4 (Exchangeablity of the Newton potential operator with any partial derivative). For any continuously differentiable vector field Q ∈ C1(Rn, Rm) that decays faster than |x|−c for |x| → ∞ and c > 0, the Newton potential operator as defined by Eq. (20) can be exchanged with any partial derivative: ∂ Z Z ∂Q(x) N Q(x) = N . (44) ∂xk ∂xk

In the context of this paper, Q can be G, R, Ri j, F, f , g, r, γ, ρ, ρi j, or φ. Proof. ! ∂ Z ∂ Z Z ∂ N Q(x) = K(x, ξ)Q(ξ) dξn = Q(ξ) K(x, ξ) dξn ∂xk ∂xk Rn Rn ∂xk ! (45) Z ∂ = Q(ξ) − K(x, ξ) dξn Rn ∂ξk 10 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates using integration by parts in the k-component

Z Z ∞ ! ∞ ∂Q(ξ) = − Q(ξ)K(x, ξ) − K(x, ξ) dξk d[ξi; i , k] (46) n−1 ξ =−∞ R k ξk=−∞ ∂ξk

(for n = 1, the outer integral over R0 in the intermediate step above is omitted) and as in the ﬁrst summand Q(ξ)K(x, ξ) → 0 for ξk → ±∞ ! Z ∂Q(ξ) Z ∂Q(x) = K(x, ξ) dξn = N ∀ k. (47) Rn ∂ξk ∂xk Corollary 5. Applying Lemma 4 in each component, it follows R R N ∆Q(x) = ∆N Q(x) = Q(x) for Q ∈ C2(Rn, Rm), (48) R R 2 D N F(x) = N DF(x) for a potential F ∈ C3(Rn, R1+n ), (49) R R and D N f (x) = N D f (x) for a vector ﬁeld f ∈ C2(Rn, Rn). (50)

Proposition 6 (Operator Identity: DD f = ∆ f ). For any twice continuously diﬀerentiable vector ﬁeld f ∈ C2(Rn, Rn), the following identity holds:

DD f (x) = grad div f (x) + ROT ROT f (x) = ∆ f (x). (51)

Proof. The first part follows from the definition of D and D, and Proposition 2 and 3. By using h i Eq. (26) stating ROT R(x) = P ∂Rkm ; 1 ≤ k ≤ n , it follows: m ∂xm ! ∂ f ∂ f j DD f (x) = grad div f + ROT i − ; 1 ≤ i, j ≤ n ∂x j ∂xi  n n !   ∂ X ∂ f X ∂ ∂ f ∂ f  =  m + k − m ; 1 ≤ k ≤ n ∂x ∂x ∂x ∂x ∂x  (52) k m=1 m m=1 m m k   Xn ∂2 f  =  k ; 1 ≤ k ≤ n = ∆ f ; 1 ≤ k ≤ n = ∆ f (x).  2  k m=1 ∂xm The first two terms cancel out because of the symmetry of second derivatives (Schwarz’s theorem) and interchange of sum and derivative.

Proposition 7 (Operator Identity: (−1)n curl curl f = ROT ROT f ). For any twice continuously diﬀerentiable vector ﬁeld f ∈ C2(Rn, Rn), the following identity holds:

(−1)n curl curl f (x) = ROT ROT f (x). (53)

Proof. This follows from Proposition 6 and the well-known identity (De La Calle Ysern and Sabina De Lis, 2019, p. 522)

∆ f (x) = grad div f (x) + (−1)n curl curl f (x). (54)

11 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

Proposition 8 (Identity for Newton Potential F: DDF = ∆F). For any twice continuously differentiable vector field f ∈ C2(Rn, Rn) that decays faster than |x|−c for |x| → ∞ and c > 0, the potential R F(x) = N D f (x) as defined by Eqs. (18) and (23) satisfies the following identity: DDF(x) = ∆F(x). (55) R R Proof. Using the definition of F(x), Corollary 5 stating N D f (x) = D N f (x), Proposition 6 stating DD = ∆, the definition of φ(x) in Eq. (18), and ∆F(x) = φ(x) from Eq. (24): Z Z Z DDF(x) = DD N D f (x) = DDD N f (x) = D∆N f (x) = D f (x) = φ(x) = ∆F(x). (56)

Note: While Proposition 6 is an operator identity for any field f (x), this derivation is valid only 2 if F(x) was constructed following Eqs. (18) and (23). For a general function Q ∈ C3(Rn, R1+n ), DDQ(x) may differ from ∆Q(x). Proposition 9 (Eqs. (35–36) of the Helmholtz Decomposition Theorem: g(x) + r(x) = f (x)). For any twice continuously differentiable vector field f : Rn → Rn that decays faster than |x|−c for R |x| → ∞ and c > 0, the source potential G(x) = N div f (x) as defined by Eq. (20) and its gradient R g(x) = grad G(x), the rotation potential R(x) = N ROT f (x) as defined by Eqs. (21) and (22) and its rotation r(x) = ROT R(x), and the potential F(x) = {G(x), R(x)} as defined by Eq. (23) satisfy the following identity: g(x) + r(x) = grad G(x) + ROT R(x) = DF(x) = f (x). (57) Proof. The first equalities follow immediately from the definitions. The last can be derived starting with the definition of F(x), applying Corollary 5 and Proposition 6: Z Z Z DF(x) = D N D f (x) = DD N f (x) = ∆N f (x) = f (x). (58)

This completes the proof of the Helmholtz Decomposition Theorem.

6. Conclusions

In this paper, we have introduced differential operators ROT, ROT and the generalized derivatives D and D, such that for any twice continuously differentiable vector field f (x) in Rn that decays faster then |x|−c for x → ∞ and c > 0, a scalar source potential G(x) and an antisymmetric rotation potential n R(x) = Ri j(x) with 2 distinct entries can be calculated as convolutions of the density φ(x) = D f (x) with the fundamental solutions of the Laplace equation. The joint potential F(x) = {G(x), R(x)} is an antiderivative of f (x), such that applying the differential operator D to this potential provides a decomposition of f (x) into a rotation-free ‘gradient field’ g(x) = grad G(x) and a source-free ‘rotation field’ r(x) = ROT R(x): R Z f (x) = DF(x) = D N D f (x) = D N φ(x) = g(x) + r(x). (59)

This generalizes the Helmholtz Decomposition to Rn without the need for differential forms and the complicated operator curl and facilitates its application to high-dimensional dynamic systems. The potentials correspond to ‘antiderivatives’ of the gradient and rotation differential operators, providing a generalization of the fundamental theorem of calculus in Rn that links differential and integral calculus. 12 Glötzl, Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates

Acknowledgments

EG thanks Ulf Klein and Walter Zulehner from Johannes Kepler University Linz. OR thanks Ulrike Feudel and Jan Freund from Carl von Ossietzky University of Oldenburg, and Anja Janischewski.

References

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A. Comparison of the Helmholtz Decomposition, operators and variables in dimension n with dimension 3

Variable or operator dimension n dimension 3

source potential G(x) Φ(x) = −G(x) (60) G ∈ C3(Rn, R)

n n basic rotation potentials Ri j(x) = −R ji(x); 1 ≤ i, j ≤ n, distinct entries R1, R2, R3, = 3 distinct entries (61) 3 n 2 2 Ri j, Ri ∈ C (R , R)

rotation potential R(x) = Ri j(x); 1 ≤ i, j ≤ n n × n matrix R(x) = A(x) = [R1, R2, R3] (62) R ∈ C3(Rn, Rn2 ) n o n o potential F(x) = G(x),R(x) F(x) = G(x), R(x) (63) F ∈ C3(Rn, R1+n2 ) " # basic rotation operator ∂Ri j ∂Ri j (64) 1 n ROTi j Ri j B δik − δ jk ; 1 ≤ k ≤ n ROTi j : C (R , R) → ∂x j ∂xi C0(Rn, Rn)   " # rotation operator X X ∂Rkm  ∂R3 ∂R2 ∂R1 ∂R3 ∂R2 ∂R1 2 1   R − , − , − 1 n n ROT R B ROTi j Ri j =  ; 1 ≤ k ≤ n curl = (65) ROT R: C (R , R ) → 2  ∂x  ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 ≤ ≤ m m C0(Rn, Rn) 1 i, j n ROT R = curl R with R1 = R23; R2 = −R13; R3 = +R1,2 vector field f , gradient f (x) = g(x) + r(x) f (x) = g(x) + r(x) (66) field g and rotation field r g(x) = grad G(x); r(x) = ROT R(x) g(x) = grad G(x); r(x) = curl R(x) (67) f, g, r ∈ C2(Rn, Rn)

gradient ﬁelds are ROTg(x) = ROT grad G(x) = 0 curlg(x) = curl grad G(x) = 0 (68) rotation-free

rotational ﬁelds are div r(x) = div ROT R(x) = 0 div r(x) = div curl R(x) = 0 (69) divergence-free

derivative of a potential f (x) = DF(x) B grad G(x) + ROT R(x) f (x) = DF(x) B − grad Φ(x) + curl R(x) (70) D: C1(Rn, R1+n2 ) → C0(Rn, Rn)

scalar source density γ γ(x) = div f (x) = ∆G(x) γ(x) = div f (x) = ∆G(x) (71) γ ∈ C1(Rn, R) ! basic rotation density op. ∂ fi ∂ f j (72) 1 n n ROTi j f (x) B − ROTi j : C (R , R ) → ∂x j ∂xi C0(Rn, R) ! " # rotation density operator ∂ fi ∂ f j ∂ f3 ∂ f2 ∂ f1 ∂ f3 ∂ f2 ∂ f1 1 n n ROT f (x) B ROTi j f (x) = − ; 1 ≤ i, j ≤ n curl f = − , − , − (73) ROT: C (R , R ) → ∂x j ∂xi ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 C0(Rn, Rn2 ) basic rotation density ρi j(x) = −ρ ji(x) = ROTi j f (x); 1 ≤ i, j ≤ n, ρ1(x), ρ2(x), ρ3(x) (74) ∈ 1 n ρi j, ρi C (R , R) n n 2 distinct entries 2 = 3 distinct entries (75)

rotation density ρ ρ(x) = ρi j(x); 1 ≤ i, j ≤ n = ROT f (x) = ROT ROT R ρ(x) = ρ1, ρ2, ρ3 = curl f (x) = curl curl R(x) (76) ρ ∈ C1(Rn, Rn2 ) n × n matrix identify: ρ = −ρ2,3; ρ = +ρ1,3; ρ = −ρ1,2 1 2 3 n o n o density derivative D f (x) B div f (x), ROT f (x) D f (x) B div f (x), curl f (x) (77) D: C1(Rn, Rn) → C0(Rn, R1+n2 ) n o density φ(x) = γ(x), ρ(x) = γ(x), ρ (x) = D f (x) φ(x) = γ(x), ρ(x) = γ(x), ρ (x), ρ (x), ρ (x) = D f (x) 2 i j 1 2 3 φ ∈ C1(Rn, R1+n ) (78) Z Z Newton potential operator N n 1 γ(ξ) 3 NR 0 n G(x) = γ(x) = K(x, ξ) γ(ξ) dξ Φ(x) = dξ (79) : C (R , R) → Rn 4π R3 |x − ξ| 2 n Z Z C (R , R) N n 1 $ ρ(ξ) 3 Ri j(x) = ρi j(x) = K(x, ξ) ρi j(ξ) dξ R(x) = dξ (80) n 4π 3 |x − ξ|  R R  1 (log |x − ξ| − log |ξ|) n = 2 $ K x, ξ  2π with the kernel ( ) =  1 2−n 2−n n n  (|x − ξ| − |ξ| ) n 2, and V = π 2 /Γ + 1 the volume of a unit n-ball, V = 4π/3 n(2−n)Vn , n 2 3