Understanding Geometric Algebra for Electromagnetic Theory by John W. Arthur

By John W. Arthur

This e-book goals to disseminate geometric algebra as an easy mathematical software set for operating with and realizing classical electromagnetic concept. it really is aim readership is someone who has a few wisdom of electromagnetic thought, predominantly usual scientists and engineers who use it during their paintings, or postgraduate scholars and senior undergraduates who're trying to develop their wisdom and elevate their knowing of the topic. it truly is assumed that the reader isn't a mathematical expert and is neither conversant in geometric algebra or its program to electromagnetic concept. the fashionable strategy, geometric algebra, is the mathematical instrument set we should always all have began with and as soon as the reader has a take hold of of the topic, she or he can't fail to gain that conventional vector research is admittedly awkward or even deceptive through comparison.

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Sample text

Under multiplication, parallel vectors commute whereas orthogonal vectors anticommute. This will turn out be a recurrent theme that we try to exploit wherever possible because most of us have an intuitive idea of what perpendicular and parallel should mean. 2â•… 3D Geometric Algebraâ•…â•… 17 geometric algebra as being a completely abstract concept, we could define parallel and perpendicular from the commutation properties alone with no geometric interpretation attached. 1(b), (d), and (e). This in turn fits back in with the commutation properties.

It is appropriate, however, that we should convert this equation to the bivector form G = I G = Id × f = d ∧ f I G = Id × f = d ∧ f . For the electric dipole, we therefore have G e = p ∧ E = 21 ( pE − Ep ) , where p takes the place of d and E takes the place of f. Another way of putting this is that G e = pE 2, that is to say, the bivector part of pE. 1 is the grade filter such that when k = 2, U k returns the bivector part of any multivector U. The case of the magnetic dipole, however, is not just as straightforward.

When the order of multiplication is reversed, in one case we must have the same result, for example, u multiplied by itself, while for the other case where u ⊥ v, the change in sign between uv and vu is to be associated with the opposite orientation of the object. The basic rules imply the following: • The product of any two vectors results in a scalar plus a bivector. • u 2 , the product of any vector u with itself, is a scalar. • There is therefore no need to define a separate inner product as the basis of the metric.

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