By Andrew Seagar
This paintings offers the Clifford-Cauchy-Dirac (CCD) method for fixing difficulties concerning the scattering of electromagnetic radiation from fabrics of all kinds.
It permits someone who's to grasp innovations that bring about less complicated and extra effective recommendations to difficulties of electromagnetic scattering than are presently in use. The process is formulated when it comes to the Cauchy kernel, unmarried integrals, Clifford algebra and a whole-field method. this is often unlike many traditional recommendations which are formulated when it comes to Green's services, double integrals, vector calculus and the mixed box fundamental equation (CFIE). while those traditional options bring about an implementation utilizing the strategy of moments (MoM), the CCD approach is applied as alternating projections onto convex units in a Banach space.
The final end result is an quintessential formula that lends itself to a extra direct and effective resolution than conventionally is the case, and applies with no exception to all kinds of fabrics. On any specific desktop, it ends up in both a swifter answer for a given challenge or the facility to resolve difficulties of larger complexity. The Clifford-Cauchy-Dirac strategy deals very actual and important merits in uniformity, complexity, pace, garage, balance, consistency and accuracy.
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Extra info for Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique
Choose the second unit vector n2 = √12 [ √12 (e1 + e3 ) + e2 ] = 21 (e1 + 2e2 + e3 ). (u) = R2 (R1 (u)) = n2 n1 un1 n2 . First do inner reflection v = R1 (u) = n1 un1 . Use result v = −5e1 + 4e2 − 3e3 from example A(c) in Sect. 2. 3 Application to Bivector Fields The electromagnetic field F is cast into the √ Clifford formalism, as for Eq. 10, in √ terms of a bivector u = F = μ Hσ − i Ee0 . 19) Both the magnetic and electric parts of the field are subject to the influence of the operator.
X yz abc . . x yz abc . . x yz ABC . . X YZ ABC . . XYZ ea eb ec . . ex e y ez abc . . x yz ABC . . X Y Z ABC . . 21) in the context of Clifford algebra. The distinction is important because the range of multiplication operations which is supported by each is different. The conventional vector operator ∇ supports only scalar, dot and cross products, whereas the Clifford vector operator supports the full range of vector and Clifford multiplication and all of the variants described in Sect.
Both changing the sign of every primal unit e p and also reversing the order of the entire sequence of primal units. The effect is to multiply every component containing a unit in grade by the factor of ξ = (−1) ( +1)/2 . 2 Addition and Subtraction Addition and subtraction of Clifford numbers is straightforward. The coefficients of components with matching units are simply added or subtracted according to the normal rules for entities of their own particular types. 9) S and the difference a − b = c is: a−b = aS eS − S bS eS S (a S − b S )e S = = S S The index set S is taken to include all of the units in both a and b.