قضیه هلمهولتز

قضیه هلم‌هولتز (به انگلیسی: Helmholtz's theorem) قضیه‌ای بنیادین در در فیزیک و ریاضیات ^[۱]^[۲]، در زمینه حساب برداری، است که همچنین به نام قضیه اساسی حساب برداری نامیده می‌شود،^[۳]^[۴]^[۵]^[۶]^[۷]^[۸]^[۹] 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.^[۱۰]

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 (به انگلیسی). University of Texas at Arlington. Retrieved 2015-04-03.
Arfken, G.B.; Weber, H.J.; Harris, F.E. (2012). Mathematical Methods for Physicists: A Comprehensive Guide (به انگلیسی). Elsevier. Retrieved 2015-04-03.
Dugdale, D.; Dugdale, D.E. (1993). Essentials of Electromagnetism. Macmillan physical science (به انگلیسی). American Inst. of Physics. Retrieved 2015-04-03.

این یک مقالهٔ خرد ریاضیات است. می‌توانید با گسترش آن به ویکی‌پدیا کمک کنید.

[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.

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