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Questions tagged [geometric-algebras]

Geometric algebras are Clifford algebras over the real numbers. They are applied in geometry and theoretical physics.

Geometric algebras are Clifford algebras over the real numbers. They can be used as a tool to study vector algebra, and they can be applied to problems in geometry and theoretical physics.

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Intuition for geometric product being dot + wedge product

While I feel quite comfortable with the meaning of the dot and exterior products separately (parallelity and perpendicularity), I struggle to find meaning in the geometric product as the combination of the two given that one’s a scalar and the other…
Kevin Goodman
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Calculating the inverse of a multivector

Given a multivector, what is the easiest way to compute its inverse? To take a concrete example, consider a bivector $ B = e_1(e_2 + e_3) $. To compute $ B^{-1} $, I can use the dual of $ B $: $$ B = e_1e_2e_3e_3 + e_1e_2e_2e_3 = I(e_3-e_2) = Ib…
user997712
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How does the geometric product work? Inconsistent/circular?

I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby. They claim in chapter 4 that the geometric product $ab$ between two vectors $a$ and $b$ is defined according to the axioms i) associativity: $(ab)c = a(bc) = abc$ ii)…
user352830
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Geometric Algebra: How to prove that the Grade Projection Operator is well defined

The Grade Projection Operator $<\cdot>_r$ is widely used in Geometric Algebra to prove numerous relations and results. I'm looking for a proof that Grade Projection is a well-defined operation. To do as such it should be sufficient to show that an…
Confusus
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Is Hestenes's Geometric Algebra widely accepted?

I would like to write a paper on the fundamentals of Continuum Mechanics using the Geometric Algebra approach popularized by David Hestenes. Is Hestenes's Geometric Algebra a wide accepted theory? I'm concerned because this previous Math.SE Answer…
Roberto Dias Algarte
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General method to calculate dual of a k-vector with arbitrary metric in Geometric Algebra

I'm implementing a Geometric Algebra library where I hope to be able to elegantly handle degenerate metrics as well as the more common euclidean and non-euclidean metrics. i.e. $\ e_i e_i = \{0,1,-1\}$ For $\ e_i e_i = \{1,-1\}$ the most common…
Michael Alexander Ewert
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Geometric Algebra and the Gradient of a Vector

In geometric algebra we have the derivative by a vector of a vector field $$\nabla V=\nabla \cdot V+\nabla \wedge V$$ While in tensor analysis we have $$\nabla V=\frac{1}{n}I(n)\nabla \cdot V +\nabla \wedge V+\sigma (V)$$ where I(n) is the…
Timothy Wofford
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Reversion in Geometric Algebra- how is it computed?

In all of the sources that I have encountered, the GA reversion for $A=nm$ is defined as $\widetilde{A}= \widetilde{m}\widetilde{n}$, with $\widetilde{v}=v$ for a vector $v$. I have two issues with this: Is it necessary to decompose a multivector…
Rd Basha
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Proof that geometric product is associative

Geometric product has nice property since it is a ring and it is associative to multiplication, which is not the case for vector cross product. But besides it is an axiom for geometric product, in the process of actually defining geometric product…
ahala
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Formula (1.18) page 43 in Hestenes book "New foundations for Classical mechanics"

The formula is $(A_r\land b)\cdot C_s=A_r\cdot (b\cdot C_s)$, where $0
alexanderyaacov
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Different answers to geometric product problem

I started looking into Clifford Algebras, but I think I am conceptually missing some points. Here is how I understand the geometric product now (leaving out coordinate system independence and the generalized definitions of the products for…
jvdh
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Proof that every noninvertiable multivector has an idempotent factor.

Prove that every multivector which does not have an inverse has an idempotent for a factor. Define an idempotent as a multivector $A$ with the property that $A^2=A$ and $A \neq 1$. I can show it for specific cases, such as, $B = \beta + \mathbf…
user137731
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Is the Reversion map in Geometric Algebra well-defined?

I am studying from the book "Geometric Algebra for Physicists: by Chris Doran and Anthony Lasenby." In the book they define a map $\dagger : \mathcal G \to \mathcal G$, where $\mathcal G$ is a geometric algebra, by $(a_1\ldots…
Jason Rodriques
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Proof of bivector multiplication with reciprocal frame vector from Doran, Lasenby?

Doran and Lasenby (Geometric Algebra for Physicists) introduce the reciprocal frame vector and make the below assertion about multiplication with arbitrary bivectors (page 102, eq 4.104): $e_i e^i \cdot (a \wedge b) = e_i e^i \cdot ab - e_i e^i…
chockeyblocky
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Apparent inconsistency in geometric product associativity

With a little bit of work, I have proven to myself that the geometric product between three vectors is associative ($a$, $b$, and $c$ are 1-vectors): $$\begin{aligned}(ab)c &= a(bc) \\ &= (b \cdot c) a - (a \cdot c) b + (a \cdot b) c + a \wedge b…
taniwha
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