Andrew Seagar's Application of Geometric Algebra to Electromagnetic PDF

By Andrew Seagar

ISBN-10: 9811000883

ISBN-13: 9789811000881

ISBN-10: 9811000891

ISBN-13: 9789811000898

This paintings offers the Clifford-Cauchy-Dirac (CCD) procedure for fixing difficulties related to the scattering of electromagnetic radiation from fabrics of all kinds.

It permits a person who's to grasp ideas that bring about less complicated and extra effective ideas to difficulties of electromagnetic scattering than are at present in use. The process 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 strategies which are formulated when it comes to Green's services, double integrals, vector calculus and the mixed box critical equation (CFIE). while those traditional strategies bring about an implementation utilizing the tactic of moments (MoM), the CCD approach is applied as alternating projections onto convex units in a Banach space.

The final end result is an essential 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 computing device, it ends up in both a speedier resolution for a given challenge or the power to unravel difficulties of higher complexity. The Clifford-Cauchy-Dirac process 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

Example text

8 shows a computer implementation “cmulx” of Eq. 23 in the Reduce programming language, for multiplying two Clifford numbers “U” and “Z” of equal but arbitrary dimension. This implementation invokes a recursive call on itself, along with calls to companion routines “subx”, “addx” and “invc”, as described in appendix A, for subtraction, addition and involution. The actual multiplication is performed by the single Reduce expression “U*Z”. Multiplying Clifford numbers of different dimensions is only slightly more complicated.

See Eqs. 7. Q16. Show that the Clifford product D2 = DD produces, to within a multiplicative constant, the d’Alembertian (the scalar wave operator). 1 Structure of Clifford Numbers Clifford numbers are multi-dimensional entities which follow certain simple rules for the arithmetic operations of addition and multiplication. Each Clifford number is composed of several different components. 1 gives examples of Clifford numbers for dimensions from zero to three. The number of components in any Clifford number depends on its dimension.

The geometric behaviour of the operator is not immediately clear from Eq. 14. However, after a closer inspection (in Sect. 1) it becomes apparent that in general indeed does behave as a rotation operator, and in particular that: • the rotation is in the plane spanned by the two vectors n1 and n2 , • the angle of rotation θ is twice the smallest angle φ between n1 and n2 , and • the direction of rotation is from n1 to n2 (covering angle φ) and beyond (by another angle φ). 16 between Clifford multiplication and the vector dot and cross products.

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Application of Geometric Algebra to Electromagnetic Scattering: The Clifford-Cauchy-Dirac Technique by Andrew Seagar


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