260A_1_Dyadic Analysis and Applications _ C T Tai

260A_1_Dyadic Analysis and Applications _ C T Tai - Radio...

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Radio Science, Volume 22, Number 7, Pages 1283-1288, December 1987 Some essential formulas in dyadic analysis and their applications C. T. Tai Radiation Laboratory, Department of Electrical Engineering and Computer Sciences, University of Michigan, Ann Arbor (Received February 2, 1987' revised June 12, 1987' accepted July 7, 1987.) Some commonly used formulas in dyadic analysis are reviewed and summarized in this article. In particular, we have derived the dyadic-dyadic version of two vector-dyadic Green's theorems. The application of these theorems to electromagnetics is illustrated by two examples. 1. INTRODUCTION Dyadic analysis as a branch of applied mathemat- ics was enunciated by American physicist J. W. Gibbs after he introduced vector analysis. In his orig- inal treatise [Gibbs and Wilson, 1913], many basic definitions in dyadic analysis were given, but none of the integral theorems were discussed. The use of dyadic analysis in electromagnetics appeared in the book by Morse and Feshbach [1953] and the one by Collin [1960]. Van Bladel, to whom this article is dedicated, was the first author to provide a list of formulas in dyadic analysis in his book Electro- magnetic Fields [Van Bladel, 1964]. In view of the growing interest of applying dyadic Green's function technique to electromagnetic prob- lems in recent years, it seems desirable to supplement Van Bladel's list and collect them in one place for the convenience of people interested in this technique. In order to present a logical development of various formulas with different degree of complexity, we decide to start from the very basic definition of dyadic analysis. Some repetition of elementary ma- terial is therefore unavoidable. Finally, we introduce two dyadic-dyadic Green's theorems which have not appeared before. Two examples are given to show the application of these theorems. 2. DYADIC ALGEBRA 2,1. Dyadic functions or dyadics A vector function or a vector, F, expressed in a Cartesian coordinate system is defined by Copyright 1987 by the American Geophysical Union. Paper number 7S0595. 0048-6604/87/007S-0595508.00 3 F = • F,•, (1) i=1 where Fi, with i= 1, 2, 3, denote the three scalar components of the vector and x i denote the three unit vectors in the direction of •i, which are com- monly referred to as x, y, z. Although (1) can be used to define a vector in any orthogonal or nonorthogon- al system, we use xi in subsequent sections to denote Cartesian variables. Now we consider three distinct vector functions denoted by 3 F•= • Foi , j=1,2,3 (2) i=1 Then a dyadic function or a dyadic is defined by 3 j=l (3) where F j, with j = 1, 2, 3, are designated as the three vector components of F. If we substitute the ex- pression for Fj defined by (2) into (3), then • can be written in the form 3 3 F= • • F u •, • (4) i=1 j=l where F u are designated as the nine scalar compo- nents of F and the doublet • as the nine unit dyadics or dyads each being formed by a pair of unit
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This note was uploaded on 12/27/2011 for the course EE 260A taught by Professor Yahyarahmat-samii during the Spring '11 term at UCLA.

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260A_1_Dyadic Analysis and Applications _ C T Tai - Radio...

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