Calculating the indices of vector fields on two- and three-dimensional Euclidean space
Abstract
We propose a method to estimate the indices of zeros or defects of vector fields in 2 and 3 dimensional Euclidean spaces. The method depends on the "Law of Vector Fields" and is implemented by computers. Using this method we give examples of electrostatic vector fields (of point charges) where the index of some zero(s) is greater than 1. We cannot prove mathematically this is the case, but our method gives the index with great certainty. The existence of such a zero answers a question posed in "The Index of Vector Fields and Electrostatics" by M. Keirouz. On the 2-dimensional Euclidean plane, we prove that the index of any isolated zero of the gradient vector field of a smooth function is at most 1. Also we give several properties of the gradient fields of harmonic functions.
Degree
Ph.D.
Advisors
Gottlieb, Purdue University.
Subject Area
Mathematics
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