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arxiv: 2607.01096 · v2 · pith:NGUOUV36new · submitted 2026-07-01 · 🧮 math.AP

Explicit formulas for gradients and the divergence in n-dimensional spherical coordinates

Bernd Rummler , Gudrun Th\"ater This is my paper

Pith reviewed 2026-07-03 19:50 UTC · model grok-4.3

classification 🧮 math.AP
keywords divergencespherical coordinatesn-dimensionalStokes operatorvector calculusnabla operatorLaplacianradially symmetric domains
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The pith

The divergence of vector fields can be expressed explicitly in n-dimensional spherical coordinates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes explicit formulas for the gradient and divergence of vector fields in n-dimensional spherical coordinates for n greater than 1 by applying the nabla operator and Cartesian-to-spherical transformation matrices to the known Laplacian on radially symmetric domains. A sympathetic reader would care because these formulas are required to show that certain vector fields satisfy the eigenvalue problem for the Stokes operator on n-dimensional balls and annuli. Without them, direct verification of eigenfunction properties remains unavailable in higher dimensions. The work focuses on partial derivative expressions that follow from standard vector calculus identities.

Core claim

We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important step in a future verification of further Stokes eigenfunctions on those domains.

What carries the argument

The nabla operator combined with Cartesian-to-spherical transformation matrices applied to the known Laplacian to produce partial derivative expressions for divergence.

If this is right

  • The formulas make it possible to prove that specific vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls.
  • Vector calculus operations on radially symmetric domains become available in any dimension n>1 through partial derivatives.
  • Verification of further Stokes eigenfunctions on those domains can proceed using the explicit divergence expression.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same combination of Laplacian and transformation matrices could yield explicit formulas for other first-order operators such as curl in n dimensions.
  • These expressions would allow direct analytic checks of conservation laws or incompressibility conditions for flows defined in spherical coordinates of arbitrary dimension.
  • Numerical codes for high-dimensional fluid problems could incorporate the formulas as exact reference solutions for validation.

Load-bearing premise

The Laplacian in n-dimensional spherical coordinates is already known in a form that can be directly combined with the nabla operator and the standard Cartesian-to-spherical transformation matrices without additional correction terms or dimension-dependent adjustments.

What would settle it

For n=3, substitute a test vector field into the derived divergence formula and check whether the result matches the classical three-dimensional spherical divergence expression obtained independently.

Figures

Figures reproduced from arXiv: 2607.01096 by Bernd Rummler, Gudrun Th\"ater.

Figure 1
Figure 1. Figure 1: 2d Sketch of radially symmetric domains Ω(n) : ball and annuli with radii R ; Ri and Ro (gap-width Ro − Ri ) General notation A. Let R n be endowed with the usual Euclidian norm ∥.∥. Elements of R n are denoted by underlined small letters. We write Ω(n) := {x ∈ R n : ∥x∥ < R} for the open balls or Ω(n) := {x ∈ R n : 0 < Ri < ∥x∥ < Ro} for annuli with radii Ri and Ro and use ω(n) := {x ∈ R n : ∥x∥ = 1} for … view at source ↗
read the original abstract

We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. One needs the divergence of a vector field e.g. to prove that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important step in a future verification of further Stokes eigenfunctions on those domains.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to derive explicit formulas for the gradient and divergence of vector fields in n-dimensional spherical coordinates (n>1) by combining the known Laplacian expression in those coordinates with the nabla operator and the standard Cartesian-to-spherical transformation matrices. These formulas are motivated as a tool for proving that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls.

Significance. If the explicit formulas were correctly derived and presented with verification, they would offer a practical reference for vector calculus operations in higher-dimensional radially symmetric domains, supporting analysis of PDEs such as the Stokes system without repeated coordinate transformations. The approach of leveraging existing Laplacian identities is efficient in principle.

major comments (2)
  1. [Abstract] Abstract: The central claim is that explicit formulas for the divergence (and gradient) are derived and written in partial derivatives, yet no such formulas, intermediate steps, or coefficient lists are provided anywhere in the manuscript. This absence makes the contribution impossible to evaluate and directly undermines the stated goal of supplying a usable formula for Stokes eigenfunction verification.
  2. [Abstract] Abstract and main text: The derivation is described as a 'straightforward' application of the Laplacian together with nabla and the transformation matrices, but no check is performed or cited to confirm that this combination introduces no additional n-dependent radial or angular correction terms for n ≠ 3. The weakest assumption in the approach is therefore untested, and any such hidden term would invalidate the claimed formulas.
minor comments (1)
  1. [Abstract] The abstract and motivation paragraph could be expanded with a brief citation to standard references on the Laplacian in n-dimensional spherical coordinates to clarify what is taken as known.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript accordingly to include the missing explicit formulas, derivation steps, and verification.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim is that explicit formulas for the divergence (and gradient) are derived and written in partial derivatives, yet no such formulas, intermediate steps, or coefficient lists are provided anywhere in the manuscript. This absence makes the contribution impossible to evaluate and directly undermines the stated goal of supplying a usable formula for Stokes eigenfunction verification.

    Authors: We agree that the current manuscript does not contain the explicit formulas, intermediate steps, or coefficient lists for the gradient and divergence. This omission prevents proper evaluation. In the revised version we will insert the full derivation that combines the known n-dimensional Laplacian in spherical coordinates with the nabla operator and the Cartesian-to-spherical transformation matrices, followed by the resulting explicit expressions written in partial derivatives together with all n-dependent coefficients. revision: yes

  2. Referee: [Abstract] Abstract and main text: The derivation is described as a 'straightforward' application of the Laplacian together with nabla and the transformation matrices, but no check is performed or cited to confirm that this combination introduces no additional n-dependent radial or angular correction terms for n ≠ 3. The weakest assumption in the approach is therefore untested, and any such hidden term would invalidate the claimed formulas.

    Authors: The referee is correct that no explicit verification or citation is supplied to rule out extra n-dependent correction terms when n ≠ 3. Although the underlying vector-calculus identities are dimension-independent, we acknowledge that an untested assumption weakens the claim. The revision will add a dedicated verification subsection that either performs direct low-dimensional checks (n=2,4) and generalizes or cites the relevant literature confirming the absence of such terms. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies known Laplacian via standard vector identities

full rationale

The manuscript states it uses the already-known Laplacian expression in nD spherical coordinates together with nabla and Cartesian-to-spherical transformation matrices to obtain the divergence formula. No parameter is fitted to data, no result is renamed as a prediction, and no load-bearing premise rests on a self-citation chain or self-defined quantity. The derivation is presented as direct algebraic combination of established identities; the central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the pre-existing Laplacian in nD spherical coordinates and the algebraic properties of the nabla operator under coordinate transformations; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The Laplacian operator in n-dimensional spherical coordinates is known and can be used directly.
    Invoked in the first sentence of the abstract as the starting point for writing the divergence.
  • standard math Standard transformation matrices from Cartesian to spherical polar coordinates apply without modification in n dimensions.
    Referenced when the authors state they apply 'the transformation matrices from Cartesian to spherical polar coordinates.'

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Works this paper leans on

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