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Rotational Systems

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Field Theory Handbook

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Abstract

Each of the 21 transformations of Section II yields one or two rotational coordinate systems, obtained by twirling the plane map about an axis. A tabulation of the most interesting of these systems is given in this section.

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Bibliography

  1. Klein, F. : Vorlesungen über lineare Differentialgleichungen der zweiten Ordnung. Bearbeitet von E. Ritter. Göttingen 1894.

    Google Scholar 

Cyelides

  1. Casey, M. : On cyclides and sphero-quartics. Phil. Trans. Roy. Soc. Lond. 161, 585 (1871)

    Article  MATH  Google Scholar 

  2. Darboux, G. : Remarques sur la théorie des surfaces orthogonales. C. R. Acad. Sci., Paris 59, 240 (1864). — Sur l’application des méthodes de la physique mathématique à l’étude des corps terminés par des cyclides. C. R. Acad. Sci., Paris 83, 1037, 1099(1876).

    Google Scholar 

  3. Maxwell, J. C. : On the cyclide. Quart. J. Math. 9, ill (1868).

    Google Scholar 

Inversion

  1. CooLIDGE, J. L.: A treatise on the circle and the sphere. London: Oxford Univ. Press 1916.

    MATH  Google Scholar 

  2. Darboux, G. : Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, p. 277. Paris: Gauthier-Villars 1910.

    MATH  Google Scholar 

  3. Jeans, J. H.: Electricity and magnetism. Cambridge: Cambridge Univ. Press 1925.

    Google Scholar 

  4. Maxwell; J. C. : Electricity and magnetism. London: Oxford Univ. Press 1904.

    Google Scholar 

  5. Schmidt, H. : Die Inversion und ihre Anwendung. Miinchen: R. Oldenbourg 1950.

    Google Scholar 

  6. Thomson, W.: Extrait d’un lettre à M. Liouville. J. Math, pures appl. 10, 364 (1845). (The first use of inversion in electrostatics.) — Extraits de deux lettres adressées à M. Liouville. J. Math, pures appi. 12, 256 (1847).

    Google Scholar 

  7. Liouville, J.: Note au sujet de l’article précédent. J. Math, pures appi. 12, 265 (1847).

    Google Scholar 

  8. Thompson, W., and P. G. Tait: Treatise on natural philosophy, Part II, p. 62. Cambridge: Cambridge Univ. Press 1890.

    Google Scholar 

  9. Wangerin, A.: Théorie des Potentials und der Kugelfunktionen, Bd. II, S. 147. Berlin: Walter de Gruyter 1921.

    MATH  Google Scholar 

  10. Weber, E.: Electromagnetic fields, p. 244. New York: John Wiley & Sons 1950.

    MATH  Google Scholar 

  11. Wrinch, D. M. : Inverted prolate spheroids. Phil. Mag. 14, 1061 (1932).

    Google Scholar 

  12. Moon, P., and D. E. Spencer: Cylindrical and rotational coordinate systems. J. Franklin Inst. 252, 327 (1951). — Some coordinate systems associated with elliptic functions. J. Franklin Inst. 255, 531 (1953).

    Article  MathSciNet  Google Scholar 

Theory of a complex variable

  1. Churchill, R. V. : Complex variables and applications. New York: McGraw-Hill Book Co. 1960.

    MATH  Google Scholar 

  2. Knopp, K.: Theory of functions. New York: Dover Publications 1945.

    Google Scholar 

  3. Nehari, Z.: Conformai mapping. New York: McGraw-Hill Book Co. 1952.

    Google Scholar 

  4. Osgood, W. F.: Lehrbuch der Funktionentheorie. Leipzig: B. G. Teubner 1928.

    Google Scholar 

  5. Rothe, R., F. Ollendorff and K. Pohlhausen: Theory of functions. Cambridge, Mass.: Technology Press 1933.

    Google Scholar 

  6. Titchmarsh, E. C. : Theory of functions. London: Oxford Univ. Press 1932.

    Google Scholar 

  7. Townsend, E. J.: Functions of a complex variable. New York: Henry Holt & Co. 1915.

    MATH  Google Scholar 

  8. Walker, M. : Conjugate functions for engineers. London : Oxford Univ. Press 1933.

    Google Scholar 

Complex transformations

  1. Kober, H.: Dictionary of conformai representations. New York: Dover Publications 1952.

    Google Scholar 

  2. Love, A. E. H. : Some electrostatic distributions in two dimensions. Proc. London Math. Soc. 22, 337 (1924).

    Article  MATH  Google Scholar 

  3. Michell, J. H.: A map of the complex Z-function: a condenser problem. Messenger of Math. 23, 72 (1894).

    Google Scholar 

  4. Greenhill, G. : Theory of a stream line past a plane barrier. Advisory Comm. for Aeronautics, Report No. 19, London 1910.

    Google Scholar 

Applications of complex transformations

  1. Andronescu, P. : Das parallel- und meridianebene Feld nebst Beispielen. Arch. Elektrotechn. 14, 379 (1924).

    Article  Google Scholar 

  2. Bairstow, L. : Resistance of a cylinder moving in a viscous fluid. Phil. Trans. Roy. Soc. Lond. 223, 383 (1923).

    Article  MATH  ADS  Google Scholar 

  3. Glauert, H. : Elements of aerofoil and airscrew theory. Cambridge: Cambridge Univ. Press 1926.

    Google Scholar 

  4. GrÖsser, W. : Einige elektrostatische Probleme der Hochspannungstechnik. Arch. Elektrotechn. 25, 193 (1931).

    Article  MATH  Google Scholar 

  5. HoltzmÜller, O.: Über die logarithmische Abbildung und die aus ihr ent-springenden orthogonalen Kurvensysteme. Z. Math. Phys. 16, 269 (1871). Beiträge zur Theorie der isogonalen Verwandtschaften. Z. Math. Phys. 18, 227 (1873). — Weitere Beiträge zur Theorie der isogonalen Verwandtschaften. Z. Math. Phys. 20, 1 (1875). — Einführung in die Theorie der isogonalen Verwandtschaften. Leipzig: B. G. Teubner 1882.

    Google Scholar 

  6. Kehren, E. : Anwendung der konformen Abbildung in der Elektrostatik. Ann. d. Phys. 14, 367 (1932).

    Article  MATH  ADS  Google Scholar 

  7. Knight, R. C. : The potential of a circular cylinder between two infinite planes. Proc. London Math. Soc. 39, 272 (1933).

    Article  Google Scholar 

  8. Labus, J.: Berechnung des elektrischen Feldes von Hochspannungstransformatoren mit Hilfe der konformen Abbildung. Arch. Elektrotechn. 19, 82 (1927). — Der Potential- und Feldverlauf längs einer Transformatorwicklung. Arch. Elektrotechn. 21, 250 (1928).

    Article  Google Scholar 

  9. Levi-Civita, T.: Sopra un problema di elettrostatica che si è presentato nella consunzione dei cavi. Rend. Cire. Math. Palermo 20 (1905).

    Google Scholar 

  10. Levy, H. : Discontinuous fluid motion past a curved boundary. Proc. Roy. Soc. Lond. 92, 285 (1915).

    ADS  Google Scholar 

  11. McLachlan, N. W. : Heat conduction in elliptical cylinder and an analogous electromagnetic problem. Phil. Mag. 36, 600 (1945).

    MATH  MathSciNet  Google Scholar 

  12. Meyer, E. : Zwei Beispiele zweidimensionaler elektrostatischer Kraftlinienbilder. Math. Ann. 93, 157 (1925).

    Article  MathSciNet  Google Scholar 

  13. Morton, W. B. : The electrification of two intersecting planes. Phil. Mag. 1, 337 (1926). — On the parallel-plate condenser and other two-dimensional fields specified by elliptic functions. Phil. Mag. 2, 827 (1926).

    Google Scholar 

  14. Nicholson, J. W. : The electrification of two parallel circular disks. Phil. Trans. Roy. Soc. Lond. 224, 303 (1923/24).

    Article  ADS  Google Scholar 

  15. Page, W. M. : Some two-dimensional problems in electrostatics and hydrodynamics. Proc. London Math. Soc. 11, 313 (1913).

    Article  Google Scholar 

  16. Petersohn, H. : Zweidimensionale elektrostatische Probleme. Z. Physik 38, 727 (1926).

    Article  ADS  Google Scholar 

  17. Poole, E. G. C. : On the discontinuous motion produced in an infinite stream by two plane obstacles. Proc. London Math. Soc. 22, 425 (1924).

    Article  MATH  Google Scholar 

  18. Poritsky, H. : Field due to two equally charged parallel conducting cylinders. J. Math. Phys. 11, 213 (1932).

    Google Scholar 

  19. Richmond, H. W. : On the electrostatic field of a plane or circular grating formed of thick rounded bars. Proc. London Math. Soc. 22, 389 (1924). — Notes on the use of the Schwartz-Christoffel transformation in electrostatics (and hydrodynamics). Proc. London Math. Soc. 22, 483 (1924).

    Article  MATH  Google Scholar 

  20. ROGOWSKI, W. : Die elektrische Festigkeit am Rande des Plattenkondensators. Arch. Elektrotechn. 12, 1 (1923).

    Article  Google Scholar 

  21. Siebeck, F. H. : Über eine Gattung von Curven vierten Grades, welche mit den elliptischen Funktionen zusammenhängen. J. reine angew. Math. 57, 359 (1860).

    Article  MATH  Google Scholar 

  22. Thomson, J. J.: Recent researches in electricity and magnetism, Chap. 3. London: Oxford Univ. Press 1893.

    Google Scholar 

  23. Weber, E. : Die konforme Abbildung in der elektrischen Festigkeitslehre. Arch. Elektrotechn. 17, 174 (1926).

    Article  Google Scholar 

  24. Wright, C. E. : Note on the potential and attraction of rectangular bodies. Phil. Mag. 10, 110 (1930).

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Parry Moon

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06268, USA

    Domina Eberle Spencer

Authors
  1. Parry Moon
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  2. Domina Eberle Spencer
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© 1961 Springer-Verlag, Berlin, Heidelberg

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Moon, P., Spencer, D.E. (1961). Rotational Systems. In: Field Theory Handbook. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83243-7_4

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  • DOI: https://doi.org/10.1007/978-3-642-83243-7_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18430-0

  • Online ISBN: 978-3-642-83243-7

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