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The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere

Record Type:	Electronic resources : Monograph/item
Title/Author:	The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere
Author:	Parsley, Robert Jason.
Description:	123 p.
Notes:	Advisers: Dennis DeTurck; Herman Gluck.
Notes:	Source: Dissertation Abstracts International, Volume: 65-03, Section: B, page: 1357.
Contained By:	Dissertation Abstracts International65-03B.
Subject:	Mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3125886
ISBN:	0496731750

The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere
Parsley, Robert Jason.

The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere [electronic resource] - 123 p.

Advisers: Dennis DeTurck; Herman Gluck.

Thesis (Ph.D.)--University of Pennsylvania, 2004.

We study the generalization of the Biot-Savart law from electrodynamics in the presence of curvature. We define the integral operator BS acting on all vector fields on subdomains of the three-dimensional sphere, the set of points in R4 that are one unit away from the origin. By doing so, we establish a geometric setting for electrodynamics in positive curvature. When applied to a vector field, the Biot-Savart operator behaves like a magnetic field; we display suitable electric fields so that Maxwell's equations hold. Specifically, the Biot-Savart operator applied to a "current" V is a right inverse to curl; thus BS is important in the study of curl eigenvalue energy-minimization problems in geometry and physics. We show that the Biot-Savart operator is self-adjoint and bounded. The helicity of a vector field, a measure of the coiling of its flow, is expressed as an inner product of BS(V) with V. We find upper bounds for helicity on the three-sphere; our bounds are not sharp but we produce explicit examples within an order of magnitude. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere. Applications of the Biot-Savart operator include plasma physics, geometric knot theory, solar physics, and DNA replication.

ISBN: 0496731750Subjects--Topical Terms:

184409
Mathematics.

The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere
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500 $a Advisers: Dennis DeTurck; Herman Gluck.
500 $a Source: Dissertation Abstracts International, Volume: 65-03, Section: B, page: 1357.
502 $a Thesis (Ph.D.)--University of Pennsylvania, 2004.
520 # $a We study the generalization of the Biot-Savart law from electrodynamics in the presence of curvature. We define the integral operator BS acting on all vector fields on subdomains of the three-dimensional sphere, the set of points in R4 that are one unit away from the origin. By doing so, we establish a geometric setting for electrodynamics in positive curvature. When applied to a vector field, the Biot-Savart operator behaves like a magnetic field; we display suitable electric fields so that Maxwell's equations hold. Specifically, the Biot-Savart operator applied to a "current" V is a right inverse to curl; thus BS is important in the study of curl eigenvalue energy-minimization problems in geometry and physics. We show that the Biot-Savart operator is self-adjoint and bounded. The helicity of a vector field, a measure of the coiling of its flow, is expressed as an inner product of BS(V) with V. We find upper bounds for helicity on the three-sphere; our bounds are not sharp but we produce explicit examples within an order of magnitude. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere. Applications of the Biot-Savart operator include plasma physics, geometric knot theory, solar physics, and DNA replication.
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