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On binary differential equations and umbilics

Published online by Cambridge University Press:  14 November 2011

J.W. Bruce  and
D.L. Fidal
J.W. Bruce
Affiliation:
Department of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, U.K.
D.L. Fidal
Affiliation:
Department of Mathematics, University of Newcastle upon Tyne, Newcastle NE1 7RU, U.K.
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Synopsis

In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation

where a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 111 , Issue 1-2 , 1989 , pp. 147 - 168
Copyright
Copyright © Royal Society of Edinburgh 1989

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