Explicit Equations to Transform from Cartesian to Elliptic Coordinates

Che Sun

doi:doi:10.11648/j.mma.20170204.12

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Explicit Equations to Transform from Cartesian to Elliptic Coordinates

Che Sun

Received: 16 January 2017 Accepted: 25 January 2017 Published: 31 October 2017

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Abstract

Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical core induces two counter-rotating irrotational eddies.

DOI	10.11648/j.mma.20170204.12
Published in	Mathematical Modelling and Applications (Volume 2, Issue 4, August 2017)
Page(s)	43-46
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Elliptic Coordinates, Cartesian Coordinates, Kirchhoff Vortex

References

[1]	Lamb, H. (1932) Hydrodynamics. Cambridge Univ. Press, 738pp.
[2]	Wang, Z. (2000), Vortex shedding and frequency selection in flapping flight. J. Fluid Mech., 410, 323-341.
[3]	Chatjigeorgiou, I., & Mavrakos, S. (2012), The analytic form of green's function in elliptic coordinates. J. Engineering Math., 72, 87-105.
[4]	Morse, P., & Feshbach, H. (1953), Methods of theoretical physics, McGraw-Hill, 1978pp.
[5]	Arfken, G. (1970), Mathematical Methods for Physicists, 2nd ed., Academic Press, 815pp.
[6]	Hazewinkel, M. (1988), Encyclopaedia of Mathematics, Springer, 488pp.
[7]	Korn G., & Korn, T. (2000), Mathematical handbook for scientists and engineers, Dover, 1152pp.
[8]	Kirchhoff, G. (1877), Vorlesungenüber mathematische Physik. Mechanik, Leipzig.
[9]	Miyazaki, T., Imai, T., & Fukumoto, Y. (1995), Three-dimensional instability of Kirchhoff’s elliptic vortex. Phys. Fluids, 7, 195-202.
[10]	Muller, B., & Yee, H. (2001), High order numerical simulation of sound generated by the Kirchhoff vortex. Comp. Visual. Sci., 4, 197-204.
[11]	Mitchell, T., & Rossi, L. (2008), The evolution of Kirchhoff elliptic vortices. Phys. Fluids, 20, 054103.

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APA Style

Che Sun. (2017). Explicit Equations to Transform from Cartesian to Elliptic Coordinates. Mathematical Modelling and Applications, 2(4), 43-46. https://doi.org/10.11648/j.mma.20170204.12

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ACS Style

Che Sun. Explicit Equations to Transform from Cartesian to Elliptic Coordinates. Math. Model. Appl. 2017, 2(4), 43-46. doi: 10.11648/j.mma.20170204.12

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AMA Style

Che Sun. Explicit Equations to Transform from Cartesian to Elliptic Coordinates. Math Model Appl. 2017;2(4):43-46. doi: 10.11648/j.mma.20170204.12

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@article{10.11648/j.mma.20170204.12,
  author = {Che Sun},
  title = {Explicit Equations to Transform from Cartesian to Elliptic Coordinates},
  journal = {Mathematical Modelling and Applications},
  volume = {2},
  number = {4},
  pages = {43-46},
  doi = {10.11648/j.mma.20170204.12},
  url = {https://doi.org/10.11648/j.mma.20170204.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20170204.12},
  abstract = {Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical core induces two counter-rotating irrotational eddies.},
 year = {2017}
}

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TY  - JOUR
T1  - Explicit Equations to Transform from Cartesian to Elliptic Coordinates
AU  - Che Sun
Y1  - 2017/10/31
PY  - 2017
N1  - https://doi.org/10.11648/j.mma.20170204.12
DO  - 10.11648/j.mma.20170204.12
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 43
EP  - 46
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20170204.12
AB  - Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical core induces two counter-rotating irrotational eddies.
VL  - 2
IS  - 4
ER  -

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Author Information

Che Sun

Institute of Oceanology, Chinese Academy of Sciences, Qingdao, China

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Plain Text BibTeX RIS

APA Style

Che Sun. (2017). Explicit Equations to Transform from Cartesian to Elliptic Coordinates. Mathematical Modelling and Applications, 2(4), 43-46. https://doi.org/10.11648/j.mma.20170204.12

Copy | Download

ACS Style

Che Sun. Explicit Equations to Transform from Cartesian to Elliptic Coordinates. Math. Model. Appl. 2017, 2(4), 43-46. doi: 10.11648/j.mma.20170204.12

Copy | Download

AMA Style

Che Sun. Explicit Equations to Transform from Cartesian to Elliptic Coordinates. Math Model Appl. 2017;2(4):43-46. doi: 10.11648/j.mma.20170204.12

Copy | Download

@article{10.11648/j.mma.20170204.12,
  author = {Che Sun},
  title = {Explicit Equations to Transform from Cartesian to Elliptic Coordinates},
  journal = {Mathematical Modelling and Applications},
  volume = {2},
  number = {4},
  pages = {43-46},
  doi = {10.11648/j.mma.20170204.12},
  url = {https://doi.org/10.11648/j.mma.20170204.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20170204.12},
  abstract = {Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical core induces two counter-rotating irrotational eddies.},
 year = {2017}
}

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TY  - JOUR
T1  - Explicit Equations to Transform from Cartesian to Elliptic Coordinates
AU  - Che Sun
Y1  - 2017/10/31
PY  - 2017
N1  - https://doi.org/10.11648/j.mma.20170204.12
DO  - 10.11648/j.mma.20170204.12
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 43
EP  - 46
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20170204.12
AB  - Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical core induces two counter-rotating irrotational eddies.
VL  - 2
IS  - 4
ER  -

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