volume | PIER Journals

2007-01-11

A Rigorous and Completed Statement on Helmholtz Theorem

Yong Feng Gui,

Dr. Yong Feng Gui

East China Research Institute of Electronic Engineering

China

Homepage

Wen-Bin Dou

Dr. Wen-Bin Dou

Southeast University

China

Homepage

, Vol. 69, 287-304, 2007

doi:10.2528/PIER06123101

Abstract

There are some limitations on the statement of classic Helmholtz theorem although it has abroad application. Actually, it only applies to simply connected domain with single boundary surface and does not provide any conclusion about the domain where discontinuities of field function exist. However, discontinuity is often encountered in practice, for example, the location of surface sources or interface of two kinds of medium. Meanwhile, most existing versions of Helmholtz theorem are imprecise and imperfect to some extent. This paper not only tries to present a precise statement and rigorous proof on classic Helmholtz theorem with the accuracy of mathematical language and logical strictness, but also generalizes it to the case of multiply connected domain and obtains a generalized Helmholtz theorem in the sense of Lebesgue measure and Lebesgue integral defined on three-dimensional Euclidean space. Meanwhile, our proof and reasoning are more sufficient and perfect. As an important application of the generalized Helmholtz theorem, the concepts of irrotational and solenoidal vector function are emphasized. The generalized Helmholtz theorem and the present conclusion should have important reference value in disciplines including vector analysis such as electromagnetics.

Citation

Copy Export

Yong Feng Gui, and Wen-Bin Dou, "A Rigorous and Completed Statement on Helmholtz Theorem," , Vol. 69, 287-304, 2007.
doi:10.2528/PIER06123101

References

1. Jackson, J. D., Classical Electrodynamics, 2nd edition, 1975.

2. Stratton, J. A., Electromagnetic Theory, John Wiley & Sons, 1941.

3. Sommerfeld, A., Electromagnetic Theory, Academic Press, 1952.

4. Collin, R. E., Field Theory of Guided Waves, Mc Graw-Hill Book Co., 1960.

5. Chew, W. C., Waves and Fields in Inhomogenous Media, Van Nostrand Teinhold, 1990.

6. Cheng, D. K., Field and Wave Electromagnetics, Addision-Wesley Publishing Company. Inc., 1983.

7. Kong, J. A., Electromagnetic Wave Theory, 2nd edition, 1990.

8. Lin, W. G., Microwave Theory and Techniques, 671, 1979.

9. Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, Part II, 1953.

10. Courant, R., Methods of Mathematical Physics, Vol. 1, Vol. 1, 1953.

11. Landau, L. D. and E. M. Lifshitz, The Classical Theory of Fields, Pergamon Press, 1962.

12. Song, W., Dyadic Green Function and Operator Theory of Electromagnetic Field, Science and technique University of China Press, 1991.

13. Rudin, W., Functional Analysis, McGraw-Hill, 1973.

14. Conway, J. B., A Course in Functional Analysis, Springer-Verlag, 1985.

15. Kreyszig, E., Introductory Functional Analysis with Application, John Wiley & Sons, 1978.

16. Tuma, J. J. and R. A.Walsh, Engineering Mathematics Handbook, 4th edition, 1998.

17. Zhou, X. L., "On independence, completeness of Maxwells equations and uniqueness theorems in electromagnetics," Progress In Electromagnetics Research, Vol. 64, 117-134, 2006.
doi:10.2528/PIER06061302

18. Zhou, X. L., "On uniqueness theorem of a vector function," Progress In Electromagnetics Research, Vol. 65, 93-102, 2006.
doi:10.2528/PIER06081202

19. Zhou, X. L., "On Helmhotlz's theorem and its interpretations," Accepted for publication in Journal of Electromagnetic Waves and Application, Vol. 21, No. 4, 471-483, 2007.