References analysis of the Wikipedia article "Helmholtz decomposition" in the English version

arxiv.org

Erhard Glötzl, Oliver Richters: Helmholtz decomposition and potential functions for n-dimensional analytic vector fields. In: Journal of Mathematical Analysis and Applications 525(2), 127138, 2023, doi:10.1016/j.jmaa.2023.127138, arXiv:2102.09556v3. Mathematica worksheet at doi:10.5281/zenodo.7512798.

A. M. Stewart: Longitudinal and transverse components of a vector field. In: Sri Lankan Journal of Physics 12, pp. 33–42, 2011, doi:10.4038/sljp.v12i0.3504 arXiv:0801.0335

Erhard Glötzl, Oliver Richters: Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates. 2020, arXiv:2012.13157.

digizeitschriften.de

Hermann von Helmholtz: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. In: Journal für die reine und angewandte Mathematik 55, 1858, pp. 25–55, doi:10.1515/crll.1858.55.25 (sub.uni-goettingen.de, digizeitschriften.de). On page 38, the components of the fluid's velocity (u, v, w) are expressed in terms of the gradient of a scalar potential P and the curl of a vector potential (L, M, N).

doi.org

George Gabriel Stokes: On the Dynamical Theory of Diffraction. In: Transactions of the Cambridge Philosophical Society 9, 1849, pp. 1–62. doi:10.1017/cbo9780511702259.015, see pp. 9–10.

Alp Kustepeli: On the Helmholtz Theorem and Its Generalization for Multi-Layers. In: Electromagnetics 36.3, 2016, pp. 135–148, doi:10.1080/02726343.2016.1149755.

D. Petrascheck, R. Folk: Helmholtz decomposition theorem and Blumenthal’s extension by regularization. In: Condensed Matter Physics 20(1), 13002, 2017, doi:10.5488/CMP.20.13002.

Wolfgang Sprössig: On Helmholtz decompositions and their generalizations – An overview. In: Mathematical Methods in the Applied Sciences 33.4, 2009, pp. 374–383, doi:10.1002/mma.1212.

Harsh Bhatia, Gregory Norgard, Valerio Pascucci, Peer-Timo Bremer: The Helmholtz-Hodge Decomposition – A Survey. In: IEEE Transactions on Visualization and Computer Graphics 19.8, 2013, pp. 1386–1404, doi:10.1109/tvcg.2012.316.

Dietmar Petrascheck: The Helmholtz decomposition revisited. In: European Journal of Physics 37.1, 2015, Artikel 015201, doi:10.1088/0143-0807/37/1/015201.

Cherif Amrouche, Christine Bernardi, Monique Dauge, Vivette Girault: Vector potentials in three dimensional non-smooth domains. In: Mathematical Methods in the Applied Sciences 21(9), 1998, pp. 823–864, doi:10.1002/(sici)1099-1476(199806)21:9<823::aid-mma976>3.0.co;2-b, Bibcode:/abstract 1998MMAS...21..823A .

A. M. Stewart: Longitudinal and transverse components of a vector field. In: Sri Lankan Journal of Physics 12, pp. 33–42, 2011, doi:10.4038/sljp.v12i0.3504 arXiv:0801.0335

Frank W. Warner: The Hodge Theorem. In: Foundations of Differentiable Manifolds and Lie Groups. (= Graduate Texts in Mathematics 94). Springer, New York 1983, doi:10.1007/978-1-4757-1799-0_6.

Cantarella, Jason; DeTurck, Dennis; Gluck, Herman (2002). "Vector Calculus and the Topology of Domains in 3-Space". The American Mathematical Monthly. 109 (5): 409–442. doi:10.2307/2695643. JSTOR 2695643.

R. Douglas Gregory: Helmholtz's Theorem when the domain is Infinite and when the field has singular points. In: The Quarterly Journal of Mechanics and Applied Mathematics 49.3, 1996, pp. 439–450, doi:10.1093/qjmam/49.3.439.

Otto Blumenthal: Über die Zerlegung unendlicher Vektorfelder. In: Mathematische Annalen 61.2, 1905, pp. 235–250, doi:10.1007/BF01457564.

Morton E. Gurtin: On Helmholtz’s theorem and the completeness of the Papkovich-Neuber stress functions for infinite domains. In: Archive for Rational Mechanics and Analysis 9.1, 1962, pp. 225–233, doi:10.1007/BF00253346.

Sheldon Axler, Paul Bourdon, Wade Ramey: Bounded Harmonic Functions. In: Harmonic Function Theory (= Graduate Texts in Mathematics 137). Springer, New York 1992, pp. 31–44, doi:10.1007/0-387-21527-1_2.

Alexandre J. Chorin, Jerrold E. Marsden: A Mathematical Introduction to Fluid Mechanics (= Texts in Applied Mathematics 4). Springer US, New York 1990, doi:10.1007/978-1-4684-0364-0.

Tomoharu Suda: Construction of Lyapunov functions using Helmholtz–Hodge decomposition. In: Discrete & Continuous Dynamical Systems – A 39.5, 2019, pp. 2437–2454, doi:10.3934/dcds.2019103.

Tomoharu Suda: Application of Helmholtz–Hodge decomposition to the study of certain vector fields. In: Journal of Physics A: Mathematical and Theoretical 53.37, 2020, pp. 375703. doi:10.1088/1751-8121/aba657.

Joseph Xu Zhou, M. D. S. Aliyu, Erik Aurell, Sui Huang: Quasi-potential landscape in complex multi-stable systems. In: Journal of the Royal Society Interface 9.77, 2012, pp. 3539–3553, doi:10.1098/rsif.2012.0434.

Edward N. Lorenz: Deterministic Nonperiodic Flow. In: Journal of the Atmospheric Sciences 20.2, 1963, pp. 130–141, doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe: Strange Attractors: The Locus of Chaos. In: Chaos and Fractals. Springer, New York, pp. 655–768. doi:10.1007/978-1-4757-4740-9_13.

Armando Manduca: MR elastography: Principles, guidelines, and terminology. In: Magnetic Resonance in Medicine, 2021, doi:10.1002/mrm.28627 PMID 33296103.

Hersh Bhatia, Valerio Pascucci, Peer-Timo Bremer: The Natural Helmholtz-Hodge Decomposition for Open-Boundary Flow Analysis. In: IEEE Transactions on Visualization and Computer Graphics 20.11, Nov. 2014, pp. 1566–1578, Nov. 2014, doi:10.1109/TVCG.2014.2312012.

jstor.org

Ton Tran-Cong: On Helmholtz’s Decomposition Theorem and Poissons’s Equation with an Infinite Domain. In: Quarterly of Applied Mathematics 51.1, 1993, pp. 23–35, JSTOR 43637902.

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Helmholtz decomposition (English Wikipedia)

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