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Īnyway, geometric algebra is not some esoteric secret. But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. Geometric algebra is, as the article points out, a more powerful version of the usual vector notation.
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There are three standard notation methods in physics: vectors, tensors, and differential forms. In the same way that Newton's notation for Calculus is no longer the dominant one, or that Maxwell's actual equations for Electromagnetism have been replaced by Heaviside's, textbooks will change because a better system has come along. > I firmly believe that in 100 years, Geometric Algebra will be the dominant way of introducing students to mathematical physics. The ideas herein are nothing short of revolutionary. >In this post we will re-invent a form of math that is far superior to the one you learned in school. Math students looking for a math-style introduction to geometric algebra should try Chisolm (2012) What you want is Clifford algebra (which Clifford himself, and later Hestenes, call “geometric algebra”) and then geometric calculus, which can be used on arbitrary manifolds, in non-metrical contexts, etc.īasic geometric algebra should be taught to advanced high school students and all undergraduates studying any technical subject. Division algebras, field extensions, and galois theory (per se) are not the tools to use for studying arbitrary-dimensional geometry. It is unexceptional (indeed, expected) to get through an American undergraduate science or engineering degree without ever taking an abstract algebra course (much less the 2+ apparently expected of German pure math students).īut in any event, the top post here by Garlef is barking up the wrong tree. The author went to MIT, worked for NASA, and now lives in San Francisco I think using the language of geometric algebras / clifford algebras in physics as this article does versus the more traditional language is just a matter of taste. but that's not what this article is about. which is an entirely different part of maths. In fact it could be considered a "quantization" of the exterior algebra. but I think if you are over something other than R you're really just talking about a field extension)Ĭlifford algebras are a sort of generalization of the exterior algebra one would have encountered in differential geometry and other spaces. I guess the quaternions are an interested example in the non-algebraically closed case. I think the author is interested in clifford algebras, or more specifically geometric algebras, rather than division algebras.ĭivision algebras tend to be quite boring (if they are finite then they are just a finite field if they are finite dimensional over an algebraically closed field then they are just the field itself.