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The Biot-Savart operator and electro...
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Parsley, Robert Jason.
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The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere.
紀錄類型:
書目-電子資源 : Monograph/item
正題名/作者:
The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere./
作者:
Parsley, Robert Jason.
面頁冊數:
123 p.
附註:
Source: Dissertation Abstracts International, Volume: 65-03, Section: B, page: 1357.
Contained By:
Dissertation Abstracts International65-03B.
標題:
Mathematics. -
電子資源:
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.
- 123 p.
Source: Dissertation Abstracts International, Volume: 65-03, Section: B, page: 1357.
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:
515831
Mathematics.
The Biot-Savart operator and electrodynamics on bounded subdomains of the three-sphere.
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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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http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3125886
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