قضیه هلمهولتز

قضیه هلم‌هولتز (به انگلیسی: Helmholtz's theorem) قضیه‌ای بنیادین در در فیزیک و ریاضیات [1][2]، در زمینه حساب برداری، است که همچینی به نام قضیه اساسی حساب برداری نامیده می‌شود،[3][4][5][6][7][8][9] بیان می‌کند که هر میدان برداری صاف و به سرعت محو شونده‌ای در فضای سه بعدی می تواند به مجموع میدان غیرگردشی (بی کرل) و میدان سلونوئیدی (بی دیورژانس) تقسیم شود؛ که این اصل بنام تجزیه هلمهولتز یا تمثیل هلمهولتز شناخته می‌شود. نام این اصل به افتخار هرمان فون هلمهولتز می‌باشد.[10]

یک فضای برداری غیر چرخشی دارای یک پتانسیل اسکالر و یک پتانسیل برداری می‌باشد، تجزیه هلمهولتز بیان می‌کند که هر میدان برداری را (که شرایط صافی و محو شوندگی را ارضا کند) می‌توان به مجموع $-\operatorname {grad} \Phi +\operatorname {curl} \mathbf {A}$ تجزیه کرد، که در آن $Φ$ میدان اسکالر و $A$ میدان برداری می‌باشند.

جستارهای وابسته

هرمان فون هلمهولتز

پانویس

On Helmholtz's Theorem in Finite Regions. By Jean Bladel. Midwestern Universities Research Association, 1958.
Hermann von Helmholtz. Clarendon Press, 1906. By Leo Koenigsberger. p357
An Elementary Course in the Integral Calculus. By Daniel Alexander Murray. American Book Company, 1898. p8.
J. W. Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, page 237, link from Internet Archive
Electromagnetic theory, Volume 1. By Oliver Heaviside. "The Electrician" printing and publishing company, limited, 1893.
Elements of the differential calculus. By Wesley Stoker Barker Woolhouse. Weale, 1854.
An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions. By William Woolsey Johnson. John Wiley & Sons, 1881.
See also: Method of Fluxions.
Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205.
See also: Green's Theorem.
A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922.
See:
- H. Helmholtz (1858) "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" (On integrals of the hydrodynamic equations which correspond to vortex motions), Journal für die reine und angewandte Mathematik, 55: 25–55. 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).
- However, Helmholtz was largely anticipated by George Stokes in his paper: G. G. Stokes (presented: 1849 ; published: 1856) "On the dynamical theory of diffraction," Transactions of the Cambridge Philosophical Society, vol. 9, part I, pages 1–62; see pages 9–10.

منابع

Hsiao, H.Y.S. (2008). Helmholtz's Theorem Based Parametric Non-rigid Image Registration. University of Texas at Arlington. ISBN 978-0-549-94225-2. Retrieved ۲۰۱۵-۰۴-۰۳.
Arfken, G.B.; Weber, H.J.; Harris, F.E. (2012). Mathematical Methods for Physicists: A Comprehensive Guide. Elsevier. ISBN 978-0-12-384654-9. Retrieved ۲۰۱۵-۰۴-۰۳.
Dugdale, D.; Dugdale, D.E. (1993). Essentials of Electromagnetism. Macmillan physical science. American Inst. of Physics. ISBN 978-1-56396-253-0. Retrieved ۲۰۱۵-۰۴-۰۳.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] On Helmholtz's Theorem in Finite Regions. By Jean Bladel. Midwestern Universities Research Association, 1958.

[2] Hermann von Helmholtz. Clarendon Press, 1906. By Leo Koenigsberger. p357

[3] An Elementary Course in the Integral Calculus. By Daniel Alexander Murray. American Book Company, 1898. p8.

[4] J. W. Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, page 237, link from Internet Archive

[5] Electromagnetic theory, Volume 1. By Oliver Heaviside. "The Electrician" printing and publishing company, limited, 1893.

[6] Elements of the differential calculus. By Wesley Stoker Barker Woolhouse. Weale, 1854.

[7] An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions. By William Woolsey Johnson. John Wiley & Sons, 1881.
See also: Method of Fluxions.

[8] Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205.
See also: Green's Theorem.

[9] A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922.

[10] See:
H. Helmholtz (1858) "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" (On integrals of the hydrodynamic equations which correspond to vortex motions), Journal für die reine und angewandte Mathematik, 55: 25–55. 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).

However, Helmholtz was largely anticipated by George Stokes in his paper: G. G. Stokes (presented: 1849 ; published: 1856) "On the dynamical theory of diffraction," Transactions of the Cambridge Philosophical Society, vol. 9, part I, pages 1–62; see pages 9–10.

[11] H. Helmholtz (1858) "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" (On integrals of the hydrodynamic equations which correspond to vortex motions), Journal für die reine und angewandte Mathematik, 55: 25–55. 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).

[12] However, Helmholtz was largely anticipated by George Stokes in his paper: G. G. Stokes (presented: 1849 ; published: 1856) "On the dynamical theory of diffraction," Transactions of the Cambridge Philosophical Society, vol. 9, part I, pages 1–62; see pages 9–10.