Multivector Differential Calculus

TL;DR

This paper introduces the fundamentals of multivector differential calculus within geometric calculus, providing explicit derivations, rules, and concepts to serve as a detailed reference for mathematical and engineering applications.

Contribution

It systematically develops the multivector differential calculus, including derivatives, adjoints, and factorization, with explicit proofs and elementary explanations.

Findings

Derived basic rules of multivector differentiation

Introduced multivector derivatives and adjoints

Discussed factorization and simplicial variables

Abstract

Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential calculus part of geometric calculus. The multivector differential is introduced, followed by the multivector derivative and the adjoint of multivector functions. The basic rules of multivector differentiation are derived explicitly, as well as a variety of basic multivector derivatives. Finally factorization, which relates functions of vector variables and multivector variables is discussed, and the concepts of both simplicial variables and derivatives are explained. Everything is proven explicitly in a very elementary level step by step approach. The paper is thus intended to serve as reference material, providing a number of details, which are usually skipped in more advanced discussions of the subject matter. The arrangement of the material closely follows {chapter 2 of D. Hestenes, G. Sobczyk, Clifford Algebra to Geometric Calculus, Kluwer, Dordrecht, 1999}.