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I am progressing through a self-study of differential forms. Progress has recently slowed due to confusion with geometric algebras where an exterior product is defined on vectors where the result of a wedge product of two 1-vectors is a 2-vector also called a bivector also called a 2-blade.

This is very similar to differential forms where the exterior product of two 1-forms is a 2-form --- and so on and so forth for k-blades and k-forms in general.

Now, on the surface so far my guess is that this k-blade space is dual to the k-form space in the manner that 1-vectors are dual to 1-forms. But, I am not confident in that description as I can't find any resources that own up to that connection -- at least so far. I have Macdonald's "Linear and Geometric Algebra" text and it does not even hint at a dual relationship with differential forms. Also, I don't quite understand the geometric product versus the exterior product yet and that may have something to do with my confusion.

Can someone point me in a direction that explains the relationship between k-forms and k-blades, if one exists?

Are there resources or texts that discuss both or are these two topics totally independent of each other?

Ted Shifrin
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K7PEH
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  • the connections between Clifford algebras and exterior algebras is discussed in the book An Introduction to Clifford Algebras and Spinors of Jayme Vaz and Roldao da Rocha –  May 29 '22 at 06:38
  • @Masacroso -- Thanks for the book suggestion. Just looked it over on Amazon and placed an order. – K7PEH May 29 '22 at 16:25
  • This answer: https://math.stackexchange.com/questions/4407545/geometric-algebra-or-differential-forms-for-electromagnetism/4409471#4409471

    discusses, very briefly, the isomorphism between the differential forms d operator and it's equivalent in geometric algebra ($\nabla \wedge$).

     – Peeter Joot May 29 '22 at 19:58
  • @PeeterJoot --- Thanks for the pointer. Your answer is long and detailed and I have read only parts closely and other parts quite briefly so far. However, a question: You seem to hint that the Hodge Dual (star) operator could be different with forms from the multivectors of GA. Or, were you merely suggesting that the multivectors might have different sign conventions than forms. – K7PEH May 30 '22 at 23:52
  • Duality with GA uses different sign conventions (author dependent), whereas the Hodge dual in differential forms has a specific sign. I haven't studied enough differential forms to understand what the exact mapping between the two is. – Peeter Joot May 31 '22 at 04:01

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