On the Moisil-Theodoresco Operator in Orthogonal Curvilinear Coordinates
The action of the Moisil-Theodoresco operator over a quaternionic valued function defined on R 3 (sum of a scalar and a vector field) in Cartesian coordinates is generally well understood. However this is not the case for any orthogonal curvilinear coordinate system. This paper sheds some new light...
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| Abstract | The action of the Moisil-Theodoresco operator over a quaternionic valued function defined on
R
3
(sum of a scalar and a vector field) in Cartesian coordinates is generally well understood. However this is not the case for any orthogonal curvilinear coordinate system. This paper sheds some new light on the technical aspect of the subject. Moreover, we introduce a notion of quaternionic Laplace operator acting on a quaternionic valued function from which one can recover both scalar and vector Laplacians in the vector analysis context. |
|---|---|
| AbstractList | The action of the Moisil-Theodoresco operator over a quaternionic valued function defined on
R
3
(sum of a scalar and a vector field) in Cartesian coordinates is generally well understood. However this is not the case for any orthogonal curvilinear coordinate system. This paper sheds some new light on the technical aspect of the subject. Moreover, we introduce a notion of quaternionic Laplace operator acting on a quaternionic valued function from which one can recover both scalar and vector Laplacians in the vector analysis context. The action of the Moisil-Theodoresco operator over a quaternionic valued function defined on R3 (sum of a scalar and a vector field) in Cartesian coordinates is generally well understood. However this is not the case for any orthogonal curvilinear coordinate system. This paper sheds some new light on the technical aspect of the subject. Moreover, we introduce a notion of quaternionic Laplace operator acting on a quaternionic valued function from which one can recover both scalar and vector Laplacians in the vector analysis context. |
| Author | Bory Reyes, J. Pérez-de la Rosa, M. A. |
| Author_xml | – sequence: 1 givenname: J. surname: Bory Reyes fullname: Bory Reyes, J. organization: ESIME-Zacatenco, Instituto Politécnico Nacional – sequence: 2 givenname: M. A. surname: Pérez-de la Rosa fullname: Pérez-de la Rosa, M. A. email: perezmaths@gmail.com, marco.perez@udlap.mx organization: Department of Actuarial Sciences, Physics and Mathematics, Universidad de las Américas Puebla |
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| Keywords | Laplace operator Primary 30G35 Moisil-Theodoresco operator hyperholomorphic functions orthogonal curvilinear coordinates Secondary 35J05 |
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| References | MoonPSpencerDEThe meaning of the vector LaplacianJ. Franklin Inst.195325665515585803810.1016/0016-0032(53)91160-0 Safarov, D.K.: On well-posedness of problems for nonclassical systems of equations. Complex methods for partial differential equations (Ankara, 1998), pp. 97–102, Int. Soc. Anal. Appl. Comput., 6, Kluwer Acad. Publ., Dordrecht (1999) Moreno, G.A., Moreno, G.T., Abreu, B.R., Bory, R.J.: Inframonogenic functions and their applications in 3-dimensional elasticity theory. Math. Methods Appl. Sci. 1–10 (2018). https://doi.org/10.1002/mma.4850 (2018) RedzicDVThe operator ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} in orthogonal curvilinear coordinatesEur. J. Phys.20012259559910.1088/0143-0807/22/6/304 GrigorevMYThree-dimensional analogue of Kolosov-Muskhelishvili formulaeModern Trends Hypercompl. Anal. Trends Math.20162016203215370617610.1007/978-3-319-42529-0_11 BarberJRKlarbringASolid Mechanics and its Applications2003BerlinSpringer MikhlinSGMultidimensional Singular Integrals and Integral Equations1962MoscowFizmatgiz[in Russian] HirotaIChiyodaKVector laplacian in general curvilinear coordinatesElectron. Commun. Jpn.1982102218 GrigorevMYRegular quaternionic polynomials and their propertiesCompl. Variables Elliptic Equ.201762913431363366250910.1080/17476933.2016.1250877 GrMThéodorescoNFunctions holomorphes dans l’espaceMath. Cluj193151421590002.27401 Kennedy, W.L.: The value of curl(curlA)-grad(divA)+div(gradA)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm curl}({\rm curl} A) - {\rm grad}({\rm div} A) + {\rm div}({\rm grad} A)$$\end{document} for an absolute vector A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}. arXiv:1409.5697v1 [physics.gen-ph] 17 (2014) LaméGSur les surfaces isothermes dans les corps homogènes en èquilibre de tempratureJ. Math. Pures Appl.18372147188 Dzhuraev, A.: Methods of singular integral equations. Translated from the Russian; revised by the author. Pitman Monographs and Surveys in Pure and Applied Mathematics, 60. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, pp. xii+311 (1992) GürlebeckKSprössigWQuaternionic Analysis and Elliptic Boundary Value Problems1990BaselBirkhäuser10.1007/978-3-0348-7295-9 Grigorev, M. Y.: Three-dimensional quaternionic analogue of the Kolosov-Muskhelishvili formulae. Hypercomplex analysis: new perspectives and applications, pp. 145-166, Trends Math., Birkhäuser/Springer, Cham (2014) KravchenkoVVShapiroMIntegral Representations for Spatial Models of Mathematical Physics1996Boca RatonCRC Press0872.35001 GriffithsDJCollegeRIntroduction to Electrodynamics2013LondonPearson FüterRAnalytische Funktionen einer Quaternionen VariablenComment Math Helvetici19324920150944210.1007/BF01202702 BitsadzeAVZahaviHSome classes of partial differential equationsAdvanced Studies in Contemporary Mathematics19884New YorkGordon and Breach Science Publishersxiv+504 MacdonaldAA survey of geometric algebra and geometric calculusAdv. Appl. Clifford Algebras2017271853891361939810.1007/s00006-016-0665-y Piercey, V.I.: The Lamé and metric coefficients for curvilinear coordinates in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^3$$\end{document}, unpublished. http://odessa.phy.sdsmt.edu/~andre/ (2007). Accessed 1 Apr 2018 Tai, C.T.: A Historical Study of Vector Analysis Technical Report RL 915-The University Of Michigan Radiation Laboratory Department of Electrical Engineering and Computer Science Ann Arbor, Michigan 48109–2122 USA (1995) Moon, P., Spencer, D.E.: Field theory handbook. Including coordinate systems, differential equations and their solution. Springer, Berlin-Göttingen-Heidelberg, vol. 1961, pp. vii+236 (2012) Kravchenko, V. V.: Applied quaternionic analysis. Research and Exposition in Mathematics, 28. Heldermann Verlag, Lemgo, pp. iv+127 (2003) A Macdonald (319_CR16) 2017; 27 DJ Griffiths (319_CR6) 2013 I Hirota (319_CR11) 1982; 102 SG Mikhlin (319_CR17) 1962 DV Redzic (319_CR22) 2001; 22 MY Grigorev (319_CR9) 2017; 62 319_CR3 R Füter (319_CR4) 1932; 4 K Gürlebeck (319_CR10) 1990 M Gr (319_CR5) 1931; 5 VV Kravchenko (319_CR14) 1996 AV Bitsadze (319_CR2) 1988 319_CR7 319_CR18 JR Barber (319_CR1) 2003 P Moon (319_CR19) 1953; 256 319_CR21 319_CR20 MY Grigorev (319_CR8) 2016; 2016 319_CR13 G Lamé (319_CR15) 1837; 2 319_CR24 319_CR12 319_CR23 |
| References_xml | – reference: Kennedy, W.L.: The value of curl(curlA)-grad(divA)+div(gradA)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm curl}({\rm curl} A) - {\rm grad}({\rm div} A) + {\rm div}({\rm grad} A)$$\end{document} for an absolute vector A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A$$\end{document}. arXiv:1409.5697v1 [physics.gen-ph] 17 (2014) – reference: MikhlinSGMultidimensional Singular Integrals and Integral Equations1962MoscowFizmatgiz[in Russian] – reference: Kravchenko, V. V.: Applied quaternionic analysis. Research and Exposition in Mathematics, 28. Heldermann Verlag, Lemgo, pp. iv+127 (2003) – reference: MacdonaldAA survey of geometric algebra and geometric calculusAdv. Appl. Clifford Algebras2017271853891361939810.1007/s00006-016-0665-y – reference: MoonPSpencerDEThe meaning of the vector LaplacianJ. Franklin Inst.195325665515585803810.1016/0016-0032(53)91160-0 – reference: GürlebeckKSprössigWQuaternionic Analysis and Elliptic Boundary Value Problems1990BaselBirkhäuser10.1007/978-3-0348-7295-9 – reference: BitsadzeAVZahaviHSome classes of partial differential equationsAdvanced Studies in Contemporary Mathematics19884New YorkGordon and Breach Science Publishersxiv+504 – reference: BarberJRKlarbringASolid Mechanics and its Applications2003BerlinSpringer – reference: Grigorev, M. Y.: Three-dimensional quaternionic analogue of the Kolosov-Muskhelishvili formulae. Hypercomplex analysis: new perspectives and applications, pp. 145-166, Trends Math., Birkhäuser/Springer, Cham (2014) – reference: Dzhuraev, A.: Methods of singular integral equations. Translated from the Russian; revised by the author. Pitman Monographs and Surveys in Pure and Applied Mathematics, 60. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, pp. xii+311 (1992) – reference: Moreno, G.A., Moreno, G.T., Abreu, B.R., Bory, R.J.: Inframonogenic functions and their applications in 3-dimensional elasticity theory. Math. Methods Appl. Sci. 1–10 (2018). https://doi.org/10.1002/mma.4850 (2018) – reference: GrigorevMYThree-dimensional analogue of Kolosov-Muskhelishvili formulaeModern Trends Hypercompl. Anal. Trends Math.20162016203215370617610.1007/978-3-319-42529-0_11 – reference: HirotaIChiyodaKVector laplacian in general curvilinear coordinatesElectron. Commun. Jpn.1982102218 – reference: RedzicDVThe operator ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} in orthogonal curvilinear coordinatesEur. J. Phys.20012259559910.1088/0143-0807/22/6/304 – reference: GrigorevMYRegular quaternionic polynomials and their propertiesCompl. Variables Elliptic Equ.201762913431363366250910.1080/17476933.2016.1250877 – reference: GriffithsDJCollegeRIntroduction to Electrodynamics2013LondonPearson – reference: LaméGSur les surfaces isothermes dans les corps homogènes en èquilibre de tempratureJ. Math. Pures Appl.18372147188 – reference: Moon, P., Spencer, D.E.: Field theory handbook. Including coordinate systems, differential equations and their solution. Springer, Berlin-Göttingen-Heidelberg, vol. 1961, pp. vii+236 (2012) – reference: Tai, C.T.: A Historical Study of Vector Analysis Technical Report RL 915-The University Of Michigan Radiation Laboratory Department of Electrical Engineering and Computer Science Ann Arbor, Michigan 48109–2122 USA (1995) – reference: KravchenkoVVShapiroMIntegral Representations for Spatial Models of Mathematical Physics1996Boca RatonCRC Press0872.35001 – reference: Piercey, V.I.: The Lamé and metric coefficients for curvilinear coordinates in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}^3$$\end{document}, unpublished. http://odessa.phy.sdsmt.edu/~andre/ (2007). Accessed 1 Apr 2018 – reference: GrMThéodorescoNFunctions holomorphes dans l’espaceMath. Cluj193151421590002.27401 – reference: Safarov, D.K.: On well-posedness of problems for nonclassical systems of equations. Complex methods for partial differential equations (Ankara, 1998), pp. 97–102, Int. Soc. Anal. Appl. Comput., 6, Kluwer Acad. Publ., Dordrecht (1999) – reference: FüterRAnalytische Funktionen einer Quaternionen VariablenComment Math Helvetici19324920150944210.1007/BF01202702 – ident: 319_CR21 – start-page: xiv+504 volume-title: Advanced Studies in Contemporary Mathematics year: 1988 ident: 319_CR2 – ident: 319_CR24 – volume: 2016 start-page: 203 year: 2016 ident: 319_CR8 publication-title: Modern Trends Hypercompl. Anal. Trends Math. doi: 10.1007/978-3-319-42529-0_11 – volume: 22 start-page: 595 year: 2001 ident: 319_CR22 publication-title: Eur. J. Phys. doi: 10.1088/0143-0807/22/6/304 – ident: 319_CR7 doi: 10.1007/978-3-319-08771-9_10 – volume: 62 start-page: 1343 issue: 9 year: 2017 ident: 319_CR9 publication-title: Compl. Variables Elliptic Equ. doi: 10.1080/17476933.2016.1250877 – ident: 319_CR20 – volume: 5 start-page: 142 year: 1931 ident: 319_CR5 publication-title: Math. Cluj – volume-title: Introduction to Electrodynamics year: 2013 ident: 319_CR6 – volume: 2 start-page: 147 year: 1837 ident: 319_CR15 publication-title: J. Math. Pures Appl. – volume: 27 start-page: 853 issue: 1 year: 2017 ident: 319_CR16 publication-title: Adv. Appl. Clifford Algebras doi: 10.1007/s00006-016-0665-y – volume-title: Solid Mechanics and its Applications year: 2003 ident: 319_CR1 – volume: 102 start-page: 1 issue: 2 year: 1982 ident: 319_CR11 publication-title: Electron. Commun. Jpn. – ident: 319_CR12 – volume-title: Integral Representations for Spatial Models of Mathematical Physics year: 1996 ident: 319_CR14 – ident: 319_CR3 – ident: 319_CR13 – volume: 4 start-page: 9 year: 1932 ident: 319_CR4 publication-title: Comment Math Helvetici doi: 10.1007/BF01202702 – volume-title: Quaternionic Analysis and Elliptic Boundary Value Problems year: 1990 ident: 319_CR10 doi: 10.1007/978-3-0348-7295-9 – volume-title: Multidimensional Singular Integrals and Integral Equations year: 1962 ident: 319_CR17 – volume: 256 start-page: 551 issue: 6 year: 1953 ident: 319_CR19 publication-title: J. Franklin Inst. doi: 10.1016/0016-0032(53)91160-0 – ident: 319_CR18 doi: 10.1002/mma.4850 – ident: 319_CR23 doi: 10.1007/978-1-4613-3291-6_6 |
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3
(sum of a scalar and a vector field) in Cartesian coordinates... The action of the Moisil-Theodoresco operator over a quaternionic valued function defined on R3 (sum of a scalar and a vector field) in Cartesian coordinates... |
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| SubjectTerms | Analysis Cartesian coordinates Computational Mathematics and Numerical Analysis Fields (mathematics) Functions of a Complex Variable Mathematics Mathematics and Statistics Spherical coordinates Vector analysis |
| Title | On the Moisil-Theodoresco Operator in Orthogonal Curvilinear Coordinates |
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