Why doesn't anyone use it except gayme coders? It's way more elegant than vector calculus and even more elegant than tensors.
>>16859592You know fucking nothing about tensors. Do BASIC differential geometry and representation theory and you'll realize your retarded "Geometric Calculus" isn't able to describe anything interesting or sophisticated. It mostly just re‑brands existing structures (literally just a real Clifford algebra Cl(V,g) built from a vector space with a metric g and the geometric product), but is AWFUL at global geometry and bundles. Modern field theory and GR are formulated on manifolds with bundles, ime. principal bundles with structure group, associated vector bundles carrying representations, connections, curvature, characteristic classes, index theory, etc. Your precious little "Geometric Calculus," meanwhile, presupposes a flat vector space with metric g. Try doing some gauge theory in your "superior" framework and you'll just end up re‑inventing principal bundles and Lie algebra-valued forms inside the Clifford bundle, in a much more awkward language.And GC also tends to organize objects by grade (scalar, vector, bivector, …), which does not generally align with irreducible Lorentz reps, hide chirality and helicity behind ad hoc constructions rather than treating them as irreps, and avoid explicit use of complex reps and 2‑spinors, even though those are exactly the structures that classify relativistic fields cleanly.You're like a crow picking up trash because it's shiny, leave Hestene's cult and learn real math>b-but we have Maxwell in one equationSo do we in differential geometry you humongous faggot. Define the Dirac-Hodge/Dirac-Kähler operator [math]D:=\mathrm{d}-\delta[/math] and you get[eqn]DF=-J[/math]
>>16859670I would have kept disagreeing but you called me a bunch of names, so now I agree.
>>16859670> Your precious little "Geometric Calculus," meanwhile, presupposes a flat vector space with metric g. This is wrong and you should feel bad.[math]R^{\rho}_{\sigma\mu\nu} = R(A\wedge B) [/math]
>>16859592>We have had a thirty-eight years' war over quaternions. He had been captivated by the originality and extraordinary beauty of Hamilton's genius in this respect, and had accepted, I believe, definitely, from Hamilton to take charge of quaternions after his death, which he has most loyally executed. Times without number I offered to let quaternions into Thomson and Tait, if he could only show that in any case our work would be helped by their use. You will see that from beginning to end they were never introduced."its been a long battle- even though they were smart this math is hard, and the vector calc and its simplifying assumptions still were useful in the telegraph. Now we got computers and the difficultly of the math can be abstracted away, but there is sociological inertia, and as >>16859670 said the modernists touched it, and science happens one death at a time.
>>16859670Came here to post exactly this, word-for-word.
>>16859592It is used, albeit under the generic name of Clifford Algebra. The field is not the main tool in geometry due to its dependence on a metric, which the core of differential geometry exists independently of. To my knowledge—which may be wrong—the identification of a form with a multivector is implicitly dependent on whatever metric your clifford algebra is using, which is an analog to the usual musical isomorphism tango.Anyone with a level of maturity in this field can appreciate the namby-pamby relation between gradients and differentials, so a similarly sloppy identification between your multivectors and forms—under whatever the metric is—will be similarly viewed with disdain. Of course, the "Geometric Algebra" movement largely preys om amateurs who have a low-to-mid level of mathematical maturity, so the nuances of differential geometry—and the obvious pitfalls of using metric-dependent clifford algebra—will be lost of the people who get indoctrinated into it.
