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

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Geometric Product

I have a problem with the geometric product: In my book the unit trivector is defined like this: $(e_{1}e_{2})e_{3}=e_{1}e_{2}e_{3}$ But that would mean $(e_{1}e_{2})e_{3}= (e_{1} \wedge e_{2})\cdot ...
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derivative formula $\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} \wedge\mathbf{r}) = (n-1)\mathbf{a}$

Assume $\mathbf{r}=\mathbf{x}−\mathbf{x}′$ is the position vector in $\mathbb{R}^n$, for constant $\mathbf{a}$, we have $$\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} ...
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derivative $\nabla \frac{\mathbf{r}}{r^k}$ in the context of Geometric Calculus

Suppose $\mathbf{r} = \mathbf{x - x'}$ is the position vector in $\mathbf{R^n}$, and $r = |\mathbf{r}| = |\mathbf{x - x'}|$. Do we have $\nabla \frac{\mathbf{r}}{r^k} = \frac{n-k-1}{r^k}$ or $\nabla ...
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A question on the dual relationship between the regressive product and the exterior product

I am trying to understand the following sentence, which I came across in a book: The underlying beauty of the Ausdehnungslehre is due to this symmetry [the duality between the regressive and ...
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Can every element of a geometric algebra be represented as the product of vectors?

An element of the n-dimensional geometric algebra $\Bbb G^n$ over the reals is of the form $M = \langle M \rangle + \langle M \rangle_1 + \langle M \rangle_2 + \cdots + \langle M \rangle_n$. My ...
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Projections in Geometric Algebra

Given a geometric algebra $G$. Let $I, A,B$ be an $n,r$ and $s$ blade, respectively. Let $P(*)=(*\rfloor I) I^{-1}$ be the projection operator onto $I$. Suppose that $P(A)=A$. The problem is to ...
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How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
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28 views

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 ...
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194 views

What good is the Commutator product?

In geometric algebra, the commutator product is defined as $A \times B = \frac 1 2 (AB - BA)$. From linear algebra, I remember that the commutator of matrices is $[A, B] = AB - BA$ and the commutator ...
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38 views

Question about domains of Geometric (Clifford) Calculus functions

For those of you familiar with Geometric Calculus, and in particular, the book Vector and Geometric Calculus by Alan MacDonald, maybe you can explain something to me. On pg 16, MacDonald defines the ...
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Proving det(A) property with outermorphism definition

Let A be a square matrix. Prove that exchanging two columns of A changes the sign of $\det(A)$. **Note: I'm pretty sure this is supposed to be "adjacent columns" Source: "Linear and Geometric ...
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How is rejection defined: Question from *Linear and Geometric Algebra*

The rejection $rej_B(a) = (a \wedge B)/B$ is a linear transformation and so has an outermorphism extension. Show that the extension cannot serve as a rejection operator on $G^n$. Hint: Use the ...
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41 views

Rank and Nullity of Projection of Multivectors onto k-Blades

Describe the image and kernel of the projection function & verify the rank-nullity theorem: The projection function is defined as $P_B(M) = {(M \cdot B)}\ /\ B$ where M is a multivector in ...
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51 views

Are subspaces also subalgebras in Geometric Algebra?

Is a subspace of GA(n) closed under the geometric product? Say we let a k-blade represent a subspace of GA(n), where k < n. Does that also represent a subalgebra of GA(n)? I can see that we ...
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Geometric Algebra Question

Show that in 3D any pair of bivectors A and B have a common factor u such that A = au and B = bu. (a, b, u vectors -- au and bu are the geometric product) The only thing I can think of is to try ...
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63 views

geometric product between nonorthogonal basis and its reciprocal basis

This is related to the question of wedge product between nonorthogonal basis and its reciprocal basis in geometric algebra. If {$e_i$} is a set of basis that are not necessarily orthogonal, and ...
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wedge product between nonorthogonal basis and its reciprocal basis in geometric algebra

in the context of geometric algebra, what's the wedge product between basis and reciprocal basis? say, if {$e_i$} is a set of basis that are not necessarily orthogonal, and {$e^i$} is the ...
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what's the relationship of tensor and multivector

what's the relationship of multivector in geometric algebra and tensor? Is tensor a subset of multivector?
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What are some simple examples I can use to demonstrate the power of geometric algebra?

What are some simple examples I can use to demonstrate the power of geometric algebra over "everyday" vector algebra? An alternative way of thinking of this question might be: what example ...
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What *is* the working form of a 1-vector in geometric algebra?

Consider the geometric algebra definition of 0-vectors (scalar), 1-vectors (vector), and the inner product. Let $a$ and $b$ be 1-vectors. Then $a + b = c$, where $c$ is another 1-vector. Now, ...
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inner product of trivector and bivector in geometric algebra

Hestenes's "New Foundations for Classical Mechanics" book (page 47, 1.1c) sets a problem to show: $\begin{aligned}\left( \mathbf{a} \wedge \mathbf{b} \wedge \mathbf{c} \right) \cdot B=\mathbf{a} ...
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Clifford Algebra and Fano Plane [closed]

Thank you for reading, I'm a novice, not a mathematician by trade this question could seem very simple (or even perhaps obvious) to many of you here. I've not yet found examples of this on the web. ...
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Is Geometric Algebra isomorphic to Tensor Algebra?

Is geometric algebra (Clifford algebra) isomorphic to tensor algebra? If so, how then would one relate a unique 2-vector (this is what I'm going to call a multivector that is the sum of a scalar, ...
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Questions about the conformal model of GA

The vectors $a$,$b$, and $e$ are null vectors in the GA generated by $R^{n+1,1}$ such that $a.e=b.e=1$ where $e$ is the vector representing a point at infinity. We are told that $a$ and $b$ (and ...
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The interpretation of the inner product betweeen two vectors

I am accustomed to Euclidean vector spaces where $x.a$ (both $x$ and $a$ being in $R^2$, say) can be interpreted as the magnitude of the projection of $x$ onto $a$ (or vice versa). Recently I have ...
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Help me evaluate this geometric product

We are in the geometric algebra generated by the vector space $R^{1,1}$. Consider three vectors, $e_+, e_-, e$ where $e_+^2=1$, $ e_-^2=-1$, $e_+.e_-=0$ and $e=e_+ +e_-$. It is straightforward to ...
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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 ...
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Cross Product Component Values

When taking the cross product, the x component of the perpendicular vector is the (signed) area of the yz projection of the parallelogram spanned by the two vectors it's orthogonal to-right? And ...
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Differential operator on a manifold in Geometric Calculus

In the context of Geometric Calculus, as stated in book Clifford Algebra to Geometric Calculus (pag. 142), let $M$ be a differentiable vector manifold, $F$ be a field on $M$ and $a$ be a tangent ...
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Adjoint of a Linear Function Acting on Bivectors

I've been self studying Doran and Lasenby's Geometric Algebra for Physicists, and I'm getting stuck on a small derivation in section 4.4.3 about the adjoint of linear functions. It's not big enough to ...
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multiplication table for conformal geometric algebra?

I'm trying to find the full multiplication table for conformal geometric algebra (should be a 32 by 32 matrix). It does not seem to be available in explicit form anywhere on the web. Can anybody help ...
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Solving equation in Clifford algebra

Let $g_i$ denote the $i$'th orthonormal basis of a Clifford algebra $G_{p,q}$, then any number in Clifford domain can be represented as follows: $$a=\sum_{i=1}^{2^{p+q}}(a_ig_{i})$$ where $a_i$ is ...
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condition for a bivector to be simple

If $B$ is a bivector and $x$ is a vector, I have to prove that Bivector $B$ is simple with $x$ (i.e. there's some vector $y$ such that $B=x\wedge y$) iff $B\wedge x = 0$ Now, for necessity part: ...
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Grassman algebra and Maxima

How can I setup the Maxima atensor package for manipulating Grassman algebra ? The documentation just mentions a grassman option for initializing the atensor package (init_atensor). I can't find any ...
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Inverse of a general nonfactorizable multivector [duplicate]

For a n-dimensional geometric algebra over $\mathbb{R}$ which allows mixed signature, if B is a non-null blade the inverse of B with respect to the geometric product will be: $$B^{-1} = ...
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How come the cross product of two planes is collinear with the direction vector of the line?

If two planes intersect in a line, explain why the cross product of the normal vectors of the planes is collinear with the direction vector of the line.
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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 ...
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Does Clifford algebra depend on the topology of manifold?

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or ...
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In Geometric Algebra, how would one express the result of a tensor product in the language of GA?

Thanks for your time and effort. I appreciate your help. I'm new to geometric algebra and I get that it supersedes linear algebra. I was wondering though how I could learn to express a tensor ...
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Within Geometric Algebra, is there a way to express a linear operator within the language of geometric algebra?

Thanks for your time and effort. I appreciate your help. I'm new to geometric algebra and I get that it supersedes linear algebra. I was wondering though how I could learn to take an operator in ...
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In Geometric Algebra, is there a geometric product between matrices?

Thanks for your help in advance. I literally just started to self-study about geometric algebra. I have some coursework background in linear algebra and was trying to make an educational bridge ...
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345 views

Looking for a clear definition of the geometric product

In brief: I'm looking for a clearly-worded definition1 of the geometric product of two arbitrary multivectors in $\mathbb{G}^n$. I'm having a hard time getting my bearings in the world of ...
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Rotor Identity $ \frac{1+ba}{|a+b|} = e^{-B\theta /2} $

To prove:the identity given above where $ a, b $ are vectors, $ B $ is the unit bivector in the $ a\wedge b $ plane and $\theta $ is the angle between $ a$ and $ b$. (From "Geometric Algebra for ...
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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 = ...
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What's the motivation to add inner product and wedge product together in geometric product

I am reading some geometric algebra notes. They all started from some axioms. But I am still confused on the motivation to add inner product and wedge product together by defining $$ ab = a\cdot b + ...
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index free proof of dot product of two wedge products

I am learning geometric algebra, and meet an identy of (edited according to Andrey's comments below) $$ (a\wedge b)\cdot(c\wedge d) = (a \cdot d)(b\cdot c) - (a \cdot c)(b \cdot d)$$ as in wiki ...
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Multivectors which commute with every vector in $ \mathbb G^n $

Let $ \mathcal C $ be the set of multivectors which commute with every vector in $ \mathbb G^n $. Show that: 1) When n is even, then $ \mathcal C $ is the set of all scalars $ a $. 2) When n is odd, ...
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120 views

Vector in the line of intersection of two planes

In the context of Geometric Algebra, in $ \mathbb{R}^3 $: Let $A$ and $B$ be bivectors (representing planes). Show that $ (\langle AB \rangle_2)^* $ is a vector in the line of intersection of the two ...
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Show that $ B \mathbf I = (-1)^{k(n-1)}\mathbf I B $

where $B$ is a $k$-vector in $n$ dimensional space and $ \mathbf I $ is the pseudoscalar.
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Show that $ \mathbf u^2 \mathbf v^2 = (\mathbf u \cdot \mathbf v)^2 - (\mathbf u \wedge \mathbf v)^2 $

where $ \mathbf u $ and $ \mathbf v $ are vectors. From Linear and Geometric Algebra by Alan Macdonald.