# Tagged Questions

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

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### Which concepts in Differential Geometry can NOT be represented using Geometric Algebra?

1. It is not clear to me that linear duals, and not just Hodge duals, can be represented in geometric algebra at all. See, for example, here. Can linear duals (i.e. linear functionals) be ...
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### What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition?

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...
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### Plane containing two points $a,b \in \mathbb{S}^3$ but avoiding two other antipodal points

Let $\mathbb{S}^3$ be the three-dimensional subset of $\mathbb{R}^4$ given by $$\mathbb{S}^3 = \{ (x_1,x_2,x_3,x_4) | x_1^2 + x_2^2 +x_3^2 +x_4^2 = 1 \}.$$ I want to construct a plane through the ...
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### Geometric interpretation of multi vectors and/or geometry product

When reading introductory texts on geometric algebra, the author usually introduces two kinds of products and provides a geometric interpretation for understanding them: the dot product or inner ...
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### What is 'base space' and relationship with scalar product?

I have a very stupid and simple question, which I do not have a clear idea on. One article that I read said, "As a basic relationship in linear algebra states, the scalar product of vectors $x$ ...
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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 ...
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 ...
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 ...