Helmholtz decomposition and potential functions for n-dimensional analytic 
vector fields

Glötzl, Erhard; Richters, Oliver

doi:10.1016/j.jmaa.2023.127138

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学術論文

Helmholtz decomposition and potential functions for n-dimensional analytic vector fields

Authors

Glötzl, Erhard
External Organizations;

/persons/resource/Oliver.Richters

Richters, Oliver
Potsdam Institute for Climate Impact Research;

URL

https://doi.org/10.48550/arXiv.2102.09556
(プレプリント)

フルテキスト (公開)

28224oa.pdf
(ポストプリント), 407KB

付随資料 (公開)

There is no public supplementary material available

引用

Glötzl, E., & Richters, O. (2023). Helmholtz decomposition and potential functions for n-dimensional analytic vector fields. Journal of Mathematical Analysis and Applications, 525(2):. doi:10.1016/j.jmaa.2023.127138.

引用: https://publications.pik-potsdam.de/pubman/item/item_28224

要旨

The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in, we replace the vector potential in R³ by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n-1)/2 rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an n-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka–Volterra equations with n species.