By Andrew Seagar
This paintings provides the Clifford-Cauchy-Dirac (CCD) method for fixing difficulties related to the scattering of electromagnetic radiation from fabrics of all kinds.
It permits someone who's to grasp thoughts that result in easier and extra effective recommendations 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 method. this can be unlike many traditional options which are formulated by way of Green's capabilities, double integrals, vector calculus and the mixed box indispensable equation (CFIE). while those traditional suggestions bring about an implementation utilizing the tactic of moments (MoM), the CCD approach is carried out as alternating projections onto convex units in a Banach space.
The final end result is an imperative formula that lends itself to a extra direct and effective resolution than conventionally is the case, and applies with out exception to all kinds of fabrics. On any specific desktop, it ends up in both a swifter resolution for a given challenge or the facility to resolve difficulties of better complexity. The Clifford-Cauchy-Dirac method bargains very actual and demanding merits in uniformity, complexity, pace, garage, balance, consistency and accuracy.
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Extra resources for Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique
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. 3. See Sect. 4 for details. 7 Casting Physical Problems into Clifford Algebra For the purpose of solving physical problems with Clifford’s algebra it is necessary to cast whatever quantities are used to describe the physical phenomena involved, and the way those quantities are observed to behave and interact with one another, into a Clifford algebraic formulation.
2 shows in addition to the operation of negation, three other operations on Clifford numbers: involution, reversion and Clifford conjugation. 3. The operation of involution (denoted by the symbol ‘¬’) involves changing the sign of every primal unit e p in every evolved unit e S . The effect is to multiply every component containing a unit in grade by the factor of ξ = (−1) . 3. 3 Periodic behaviour of reversion, involution and Clifford conjugation p=1 −e S = − = ξe S e p = ξe S ξ=1 p=1 e p ξ = −1 p=1 (−e p ) p=1 e +1− p p=1 (−e +1− p ) grade Change sign Reverse order Reverse order Change sign ξ = (−1) ξ = (−1) ( −1)/2 ξ = (−1) ( +1)/2 0 1 2 3 4 5 6 7 8 9 10 11 involution + − + − + − + − + − + − reversion + + − − + + − − + + − − Cliﬀord + − − + + − − + + − − + conjugation 22 2 Numbers and Arithmetic The operation of reversion (denoted by the tilde symbol ‘ ’) involves reversing the entire sequence of primal units e p in every evolved unit e S .
4a, c. The torque is a bivector and the work is a scalar. Calculating the central product ab produces the torque in grade two of the result and produces the work in grade zero of the result. Depending on the physical situation it may be one or other of these components which is of most interest. 6. 18) Note that the involution operation (in the two rightmost columns) plays a role for the primal units e p of Clifford numbers similar to the role the complex conjugate operation plays for the imaginary unit i of complex numbers.
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