By Andrew Seagar
This paintings offers the Clifford-Cauchy-Dirac (CCD) approach for fixing difficulties related to the scattering of electromagnetic radiation from fabrics of all kinds.
It permits a person who's to grasp thoughts that result in less complicated and extra effective strategies to difficulties of electromagnetic scattering than are presently in use. The strategy is formulated when it comes to the Cauchy kernel, unmarried integrals, Clifford algebra and a whole-field method. this can be not like many traditional suggestions which are formulated by way of Green's services, double integrals, vector calculus and the mixed box crucial equation (CFIE). while those traditional strategies result in an implementation utilizing the strategy of moments (MoM), the CCD process is applied as alternating projections onto convex units in a Banach space.
The final final result is an imperative formula that lends itself to a extra direct and effective answer than conventionally is the case, and applies with no exception to every kind of fabrics. On any specific computer, it leads to both a swifter resolution for a given challenge or the facility to unravel difficulties of higher complexity. The Clifford-Cauchy-Dirac method deals very actual and important merits 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
11. Q9. Show that the two sets of quaternion units in Eqs. 11 are commutative. Q10. Calculate all sixteen units of Clifford’s four-dimensional algebra. Q11. Calculate the cross product in Cartesian components from Eq. 15 and verify from an independent source. ∂ e3 into Eq. 15 to produce an expression Q12. Substitute a = = ∂∂x e1 + ∂∂y e2 + ∂z for curl. Expand and verify from an independent source. Q13. Substitute a = into Eq. 16 to produce an expression for the Clifford product b in terms of div and curl.
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. 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.
9) Q− u = 21 (I − Q)u = 21 (ut + un − ut + un ) = un = (n · u)n The projection operators Q+ u and Q− u therefore split any Clifford vector u into normal and tangential components with respect to the surface for which n is the normal vector. 2 Numerical Examples A. Reflection (a) Reflect the vector u = 3e1 + 3e2 in the y-axis. • • • • We need a unit vector perpendicular to the y-axis. Choose the unit vector na = 1e1 + 0e2 . Then Ra (u) = na una = e1 (3e1 + 3e2 )e1 = (−3 + 3e1 e2 )e1 = −3e1 + 3e2 See Fig.