Abstract/Details

Electrostatics and the index of vector fields


1991 1991

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Abstract (summary)

Let $\varphi$ be an electrostatic potential defined by $N$ stationary charged particles in R$\sp3$, and let $N\sb\varphi$ denote its set of critical points. What can be said about $N\sb\varphi$? This question attracted the interest of many scientists including a number of mathematicians.

J. C. Maxwell tried to describe $N\sb\varphi$, but, unfortunately, he made a few errors. A. Yanushauskas showed that if $K$ is a compact subset of R$\sp3$, then $K \cap N\sb\varphi$ consists of a finite number of isolated points and a finite number of pieces of analytic curves. Then, S. S. Cairns and M. Morse proved that if $\varphi$ is nondegenerate and the total charge is not zero, then $N\sb\varphi$ consists of isolated points whose number is at least $N$ $-$ 1.

In this thesis we use the law of vector fields and some of its properties, in addition to other ideas from different branches of mathematics, to find various results. First, we note two of Maxwell's mistakes and prove a theorem about the angles of intersection of level curves. Next, we answer two questions by Morse and Cairns, and correct their mistake. Then we describe $N\varphi$ when $\varphi$ is noncompletely-degenerate, the total charge is not zero, and the particles are on a straight line. Finally, we pose the following interesting question. Is there a singular point of an electrostatic field or its restriction to a plane, whose index is equal to a given integer n? We show a singular point with index $-$2 in the xy-plane and index 0 in the space, then we make several conjectures about other integers.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
Electrostatics and the index of vector fields
Author
Keirouz, Malhab Chafic
Number of pages
75
Publication year
1991
Degree date
1991
School code
0183
Source
DAI-B 53/01, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
Advisor
Gottlieb, Daniel H.
University/institution
Purdue University
University location
United States -- Indiana
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9215577
ProQuest document ID
303927061
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/303927061/abstract
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