Articles of geometric algebras

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.

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 represented using the geometric algebra formalism? 2. It also seems like most tensors cannot be represented, see for example here. This […]

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 “geometric algebra”, even though I’m using as my guide an introductory undergraduate-level2 book (Linear and geometric algebra by Macdonald). Among the general problems that I’m […]

Are all manifolds in the usual sense also “vector manifolds”?

In geometric calculus, there is a concept of a vector manifold where the points are considered vectors in a general geometric algebra (a vector space with vector multiplication) which can then be shown to have the properties of a manifold (tangent spaces etc.). For a more precise definition of a vector manifold, see page 65 […]

Why use geometric algebra and not differential forms?

This is somewhat similar to Are Clifford algebras and differential forms equivalent frameworks for differential geometry?, but I want to restrict discussion to $\mathbb{R}^n$, not arbitrary manifolds. Moreover, I am interested specifically in whether $$(\text{differential forms on }\mathbb{R}^n\text{ + a notion of inner product defined on them}) \simeq \text{geometric algebra over }\mathbb{R}^n$$ where the isomorphism […]

Good introductory book on geometric algebra

The title of the question already says it all but I would like to add that I would really like the book to be more about geometric algebra than its applications : it should contain theorems’ proofs. Just adding that I have never taken a course on geometric algebra. I’m a 2nd year engineering student, […]

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 exact algebraic definitions or explain the relationship of everything in the list: I just want to know as much […]

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?