Differential Operators in the G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the differential operators in the G(8,2) Geometric Algebra, called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). The differential operators are three x, y, and z-direction bivector-valued differential elements and either the commutator product or the anti-commutator product for multiplication into a geometric entity that represents the function to be differentiated. The general form of a function is limited to a Darboux cyclide implicit surface function. Using the commutator product, entities representing 1st, 2nd, or 3rd order partial derivatives in x, y, and z can be produced. Using the anti-commutator product, entities representing the anti-derivation can be produced from 2-vector quadric surface and 4-vector conic section entities. An operator called the pseudo-integral is defined and has the property of raising the x, y, or z degree of a function represented by an entity, but it does not produce a true integral. The paper concludes by offering some basic relations to limited forms of vector calculus and differential equations that are limited to using Darboux cyclide implicit surface functions. An example is given of entity analysis for extracting the parameters of an ellipsoid entity using the differential operators.

Comments: 30 Pages.

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Submission history

[v1] 2015-12-13 02:52:45
[v2] 2015-12-16 23:13:56
[v3] 2015-12-18 01:51:12
[v4] 2015-12-18 20:37:40

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