Article

# Geometric Algebra

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## Abstract

This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that's strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra." My goal in these notes is to describe geometric algebra from that standpoint and illustrate its usefulness. The notes are work in progress; I'll keep adding new topics as I learn them myself.

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Fundamentos de mecánica clásica. Contenido: Orígenes del algebra geométrica; Desarrollo del algebra geométrica; Mecánica de una partícula simple; Fuerzas centrales en un sistema de dos partículas; Operadores y trasformaciones; Sistemas de muchas partículas; Mecánica de los cuerpos rígidos; Mecánica celeste; Mecánica relativista.
• Pertti Lounesto
• Clifford Algebras
• Spinors
Pertti Lounesto, Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series 286, 2nd ed. (Cambridge: Cambridge University Press, 2001).