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ⓘ Dekompozimi i Helmholcit. Ne matematike, ne degen e analizes vektoriale, Teorema e Helmholcit, e njohur gjithashtu si teorema themelore e analizes vektoriale, p ..




                                     

ⓘ Dekompozimi i Helmholcit

Ne matematike, ne degen e analizes vektoriale, Teorema e Helmholcit, e njohur gjithashtu si teorema themelore e analizes vektoriale, pohon se nje fushe vektoriale e lemuar mjaftueshem, qe zvogelohet eksponencialisht mund te ndahet ne komponente te fushes vektoriale irrotacionale dhe solenoidale.

Kjo implikon qe çdo fushe vektoriale F {\displaystyle \mathbf {F} } mund te konsiderohet si e perbere nga nje çift potencialesh: nje potencial skalar ϕ {\displaystyle \phi } dhe nje potenciali vektorial A {\displaystyle \mathbf {A} }.

Dekompozimi rezultues i Helmholcit i nje fushe vektoriale, e cila eshte dy here e diferencueshme dhe kaq e shpejte sa te veje ne zero ne infinit, e ndan fushen vektoriale ne nje shume te nje gradienti dhe te rrotacioni si me poshte:

F = − ∇ G ∇ ⋅ F + ∇ × G ∇ × F {\displaystyle \mathbf {F} =-\nabla \,{\mathcal {G}}\nabla \cdot \mathbf {F}+\nabla \times {\mathcal {G}}\nabla \times \mathbf {F}}

ku G {\displaystyle {\mathcal {G}}} tregon operatorin e potencialit Njutonian.

Neqoftese ∇ ⋅ F = 0 {\displaystyle \nabla \cdot \mathbf {F} =0}, ne themi se F {\displaystyle \mathbf {F} } eshte nje fushe solenoidale pa divergjence dhe dekompozimi i Helmholcit i F {\displaystyle \mathbf {F} } bie ne

F = ∇ × G ∇ × F = ∇ × A {\displaystyle \mathbf {F} =\nabla \times {\mathcal {G}}\nabla \times \mathbf {F}=\nabla \times \mathbf {A} }

Ne kete rast, A {\displaystyle \mathbf {A} } njihet si potenciali vektorial per F {\displaystyle \mathbf {F} }.

Njesoj, neqoftese ∇ × F = 0 {\displaystyle \nabla \times \mathbf {F} =\mathbf {0} } atehere F {\displaystyle \mathbf {F} } thuhet se eshte pa rrotacion ose irrotacionale keshtu qe dekompozimi i Helmholcit i F {\displaystyle \mathbf {F} } shnderrohet ne

F = − ∇ G ∇ ⋅ F = − ∇ ϕ. {\displaystyle \mathbf {F} =-\nabla \,{\mathcal {G}}\nabla \cdot \mathbf {F}=-\nabla \phi.}

Ne kete rast, ϕ {\displaystyle \phi } njihet si potenciali skalar per F {\displaystyle \mathbf {F} }.

Ne pergjithesi gradienti negativ i potencialit skalar barazohet me komponentin irrotacional, dhe rrotacioni i potencialit vektorial barazohet me komponentin solenoidal:

F = − ∇ ϕ + ∇ × A {\displaystyle \mathbf {F} =-\nabla \phi +\nabla \times \mathbf {A} }.
                                     

1. Referime

Referime te pergjithshme

  • George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists International Edition, 6th edition, Academic Press: San Diego 2005 pp. 95-101
  • George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th edition, Academic Press: San Diego 1995 pp. 92-93

Referime per formulimin e dobet

  • V. Girault and P.A. Raviart. Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986.
  • C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. "Vector potentials in three dimensional non-smooth domains." Mathematical Methods in the Applied Sciences, 21, 823–864, 1998.
  • R. Dautray and J L. Lions. Spectral Theory and Applications, volume 3 of Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag, 1990.

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