Sparse Representations of Clifford and Tensor algebras in Maxima

TL;DR

This paper demonstrates how Clifford algebras can be computationally implemented in Maxima, showcasing their applications in electromagnetism and comparing them with traditional tensor calculations.

Contribution

It introduces two new Maxima packages for Clifford algebra computations and applies them to electromagnetism, enhancing computational tools for these algebras.

Findings

Clifford algebra packages simplify calculations in electromagnetism.

Comparison shows advantages over traditional tensor methods.

New tools facilitate routine Clifford algebra computations.

Abstract

Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an extensive demonstration of the applications of Clifford algebras in electromagnetism using the geometric algebra G3 = Cl(3,0) as a computational model in the Maxima computer algebra system. We compare the geometric algebra-based approach with conventional symbolic tensor calculations supported by Maxima, based on the itensor package. The Clifford algebra functionality of Maxima is distributed as two new packages called clifford - for basic simplification of Clifford products, outer products, scalar products and inverses; and cliffordan - for applications of geometric calculus.