By Andrew Seagar
This paintings provides the Clifford-Cauchy-Dirac (CCD) process for fixing difficulties concerning the scattering of electromagnetic radiation from fabrics of all kinds.
It permits a person who's to grasp options that bring about easier and extra effective options to difficulties of electromagnetic scattering than are presently in use. The procedure is formulated when it comes to the Cauchy kernel, unmarried integrals, Clifford algebra and a whole-field strategy. this can be not like many traditional strategies which are formulated by way of Green's features, double integrals, vector calculus and the mixed box imperative equation (CFIE). while those traditional thoughts result in an implementation utilizing the strategy of moments (MoM), the CCD method is carried out as alternating projections onto convex units in a Banach space.
The final consequence is an vital formula that lends itself to a extra direct and effective answer than conventionally is the case, and applies with no exception to all kinds of fabrics. On any specific desktop, it leads to both a swifter resolution for a given challenge or the facility to unravel difficulties of larger complexity. The Clifford-Cauchy-Dirac strategy deals very genuine and important merits in uniformity, complexity, pace, garage, balance, consistency and accuracy.
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Additional resources for Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique
This shorter version of u can alternatively be written as u cos2 φ (as in Fig. 4) or as (v + w) cos2 φ. z u = v+w n1 ×n2 u cos2 φ +w −(v−w) sin2 φ v = −u×(n1 ×n2 ) sin1 φ y v (v−w) sin2 φ x −w v−w Fig. 4 Geometrical interpretation of rotation operator 46 3 Geometry The second term of Eq. 15 can be simplified by noting that u × (n1 × n2 ) = (v + w) × (n1 × n2 ). But w is parallel to n1 × n2 and the cross product of two parallel vectors is zero. This leaves u × (n1 × n2 ) = v × (n1 × n2 ), which represents a new vector in the plane z = 0, perpendicular to v, with length equal to that of v scaled by sin φ = |n1 × n2 |.
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. 2. (c) Reflect the vector u = 3e1 + 4e2 + 5e3 in the plane x + z = 0. • We need a unit vector perpendicular to the plane. • Choose the unit vector n = √12 (e1 + e3 ). • Then R(u) = nun = √12 (e1 + e3 )(3e1 + 4e2 + 5e3 ) √12 (e1 + e3 ) = 21 ((−3 − 5) + 4e1 e2 − 4e2 e3 + (5 − 3)e1 e3 )(e1 + e3 ) = 21 (−8 + 4e1 e2 − 4e2 e3 + 2e1 e3 )(e1 + e3 ) = 21 (−8e1 + 4e2 − 4e1 e2 e3 + 2e3 − 8e3 + 4e1 e2 e3 + 4e2 − 2e1 ) = 21 (−10e1 + 8e2 − 6e3 ) = −5e1 + 4e2 − 3e3 B.
Q17. Implement using a computer language of your choice, software to multiply two arbitrary Clifford numbers using the left inner product. A17. Check your software against question 10. Q18. Implement using a computer language of your choice, software to multiply two arbitrary Clifford numbers using the right inner product. A18. Check your software against question 11. Q19. Implement using a computer language of your choice, software to multiply two arbitrary Clifford numbers using the scalar product.