By Chen-To Tai
Unequalled in its assurance of the subject, the 1st version of GENERALIZED VECTOR AND DYADIC research helped revolutionize the remedy of boundary-value difficulties, developing itself as a vintage within the box. This increased, revised version is the main accomplished publication to be had on vector research based upon the hot technique symbolic vector. GENERALIZED VECTOR AND DYADIC research offers a copious record of vector and dyadic identities, besides quite a few varieties of Green's theorems with derivations. additionally, this version provides an ancient examine of the previous mis-understandings and contradictions that experience happened in vector research shows, furthering the reader's knowing of the subject.Sponsored by:IEEE Antennas and Propagation Society.
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Extra info for Generalized Vector and Dyadic Analysis: Applied Mathematics in Field Theory, 2nd Ed. (IEEE Press Series on Electromagnetic Wave Theory)
Sample text
In many problems, it is sometimes rather difficult to find the explicit form of F 3 unless F 1 and F2 are relatively simple functions. 26) and 3 dl = Lh; dv, u;. 27) i=1 A scalar function f is then assumed to be a function of (VI, V2, V3), and a vector function f would be a function of both the V; 's and 's, Integrals of Types I and IV can be evaluated by expressing the V; 's and h;'s in terms of a single parameter, as was done previously. The integrands of integrals of Types II, III, and V contain tt;'S that are, in general, not constant vectors, so they cannot be removed to the outside of these integrals.
00 ~v-+o 6. V ;=1 = 0, which yields a . =0. L,. 27) is a very useful theorem; it will be designated as the closed surface theorem. 28 Coordinate Systems Chap. 2 2-2 Orthogonal Curvilinear System (OCS) The GCS degenerates to the orthogonal curvilinear system (OCS) when the primary vectors are mutually perpendicular to each other. 28) where Uj denote the unit vectors along the coordinates CJ)i and hi, the metric coefficients in the OCS. For a specific system, such as the sph~rical ~oordinate system with coordinate variables R, a, cp, we use the notations R, a, and ep to denote the unit vectors in that system.
By transforming A j, Dij, and Xi into the primed functions, we find c = C' = LA~D~nx~. 102); hence we have a very useful identity: = T = D·A. 110) Hence A · Ds = b, · A. 111) 21 Sec. 96), we have A·I = I·A =A. 112) The tensor of I can be called a unit tensor, with the three diagonal terms equal to unity and the rest are null. _ There are two vector products between A and D. These products are both dyadics. ] = L DijAk(Xk x x;)Xj, i,j,k where i, j, k = 1, 2, 3 in cyclic order. 114) D;jAkX;(Xj x Xk).