By Andrew Seagar
This paintings offers the Clifford-Cauchy-Dirac (CCD) process for fixing difficulties concerning the scattering of electromagnetic radiation from fabrics of all kinds.
It permits an individual who's to grasp strategies that bring about less complicated and extra effective suggestions to difficulties of electromagnetic scattering than are presently in use. The approach is formulated by way of the Cauchy kernel, unmarried integrals, Clifford algebra and a whole-field strategy. this can be not like many traditional recommendations which are formulated when it comes to Green's services, double integrals, vector calculus and the mixed box critical equation (CFIE). while those traditional ideas result in an implementation utilizing the tactic of moments (MoM), the CCD process is carried out as alternating projections onto convex units in a Banach space.
The final end result is an crucial formula that lends itself to a extra direct and effective resolution than conventionally is the case, and applies with no exception to every kind of fabrics. On any specific desktop, it leads to both a swifter resolution for a given challenge or the facility to resolve difficulties of larger complexity. The Clifford-Cauchy-Dirac strategy deals very genuine and demanding benefits in uniformity, complexity, velocity, garage, balance, consistency and accuracy.
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Additional info for Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique
3. 4) p The x p are the coefficients, and the e p are the units, all being primal units (from grade one). The index p varies through some range of integers. For an m-dimensional vector the index plays the role of m integers, one for each of the m units therein. Integer values may well be chosen in the range from 0 to m − 1 or from 1 to m, according to preference. 5) where x p = x p−1 and e p = e p−1 . For problems in electromagnetism the unit e0 is taken to represent the dimension of time or frequency, and the units e1 , e2 , e3 are taken to represent the three orthogonal dimensions of space.
16 between Clifford multiplication and the vector dot and cross products. 15) In order to verify that Eq. 15 represents a rotation, consider now Fig. 4. 3 Rotation 45 where the v component is the projection of u onto the xy plane z = 0, and the w component is the projection of u onto the z-axis. The Cartesian coordinate system has been set in space so that the plane z = 0 coincides with the plane spanned by the two unit vectors n1 and n2 . Those unit vectors are not shown to avoid making the diagram too cluttered.
Then Ra (u) = na una = e1 (3e1 + 3e2 )e1 = (−3 + 3e1 e2 )e1 = −3e1 + 3e2 See Fig. 2. (b) Reflect the vector u = 3e1 + 3e2 in the line through the origin with slope 3/2. • We need a unit vector perpendicular to the specified line. • Choose the unit vector nb = √113 (−3e1 + 2e2 ). Fig. 2 Reflection and Projection 41 • Then Rb (u) = nb unb = √113 (−3e1 + 2e2 )(3e1 + 3e2 ) √113 (−3e1 + 2e2 ) = = = 1 ((9 − 6) + (−9 − 6)e1 e2 )(−3e1 13 1 (3 − 15e1 e2 )(−3e1 + 2e2 ) 13 1 ((−9 + 30)e1 + (6 + 45)e2 ) 13 21 e + 51 e 13 1 13 2 + 2e2 ) = • See Fig.