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Complex potential equations I. A technique for solution

Published online by Cambridge University Press:  24 October 2008

C. B. Collins
C. B. Collins
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
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Abstract

This work amalgamates and solves certain problems arising in differential equation theory and in classical differential geometry.

It describes a novel technique for solving systems of non-linear partial differential equations of the form

where f(γ) and g(γ) are arbitrarily assigned functions. The circumstances are determined under which compatible solutions exist, not only when γ is real, but also when γ is complex, and all of the corresponding solutions are found. This is done by using a geometric technique that incorporates the equipotential surfaces of constant γ. In general, these surfaces are imaginary, and a fairly extensive treatment of such surfaces in (complexified) 3-dimensional Euclidean space is included. A close association is discovered between the set of equipotential surfaces and the class of surfaces of constant radii of principal normal curvature.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 1 , July 1976 , pp. 165 - 187
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

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