DiracClifford operator amp Maxwells equations with geometric algebra

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Excerpt from • Grassmann-Hodge ★ Complement with cha... • Foundations of Differential Geometric Algebra (PDF draft, 2021) recovered and presented: • The Grassmann.jl package provides tools for doing computations based on multi-linear algebra, differential geometry, and spin groups using the extended tensor algebra known as Leibniz-Grassmann-Clifford-Hestenes geometric algebra. Combinatorial products include ∧, ∨, ⋅, *, ⋆, ', ~, d, ∂ (which are the exterior, regressive, inner, and geometric products; along with the Hodge star, adjoint, reversal, differential and boundary operators). The kernelized operations are built up from composite sparse tensor products and Hodge duality, with high dimensional support for up to 62 indices using staged caching and precompilation. Code generation enables concise yet highly extensible definitions. The DirectSum.jl multivector parametric type polymorphism is based on tangent bundle vector spaces and conformal projective geometry to make the dispatch highly extensible for many applications. Additionally, the universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built. • https://crucialflow.com • Tools built on these foundations enable computations based on multi-linear algebra and spin groups using the geometric algebra known as Grassmann algebra or Clifford algebra. This foundation is built on a direct-sum parametric type system for tangent bundles, vector spaces, and also projective and differential geometry. Geometric algebra is a mathematical foundation for differential geometry, which can be used to simplify the Maxwell equations to a single wave equation due to the geometric product. Introduction of geometric algebra to engineering science disciplines will be easier with programmable foundations.In order to devise an expressive and performance oriented language for efficient discrete differential geometric algebra with the Grassmann elements, an efficient computer algebra representation was programmed. With this unifying mathematical foundation, it is possible to improve efficiency of multi-disciplinary research using geometric tensor calculus by relying on universal mathematical principles. Tools built on universal differential geometric algebra provide a natural geometric language for the Helmholtz decomposition and Hodge-DeRahm co/homology.

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