Anonymous at Wed, 3 Jul 2024 19:59:56 UTC No. 16266490

It is cancer and always felt hand waving to me too. It may be rigorous using some measure theory shit but idk I passed my exams and I'm now done with that nonsense.

Anonymous at Wed, 3 Jul 2024 20:00:43 UTC No. 16266492

Wirtinger derivatives are uglier. Fuck complex analysis. Differentiating with respect to a 2-dimensional object is just wrong.

Anonymous at Thu, 4 Jul 2024 00:13:46 UTC No. 16266906

>>16266482
What’s the alternative to differential forms?

Anonymous at Thu, 4 Jul 2024 03:59:14 UTC No. 16267170

>>16266906
Clifford Algebra (particularly look up Geometric algebra). I recommend Doran's book on the subject. The calling card of differential forms is the fact that there are certain things like integrals that are invariant under metrics, and psueds will falsely claim that clifford algebra needs a more complicated calculation. This is false; geometric calculus and its approach to nabla let's you have all the strengths of forms with none of the weaknesses.

Anonymous at Thu, 4 Jul 2024 05:25:19 UTC No. 16267243

>>16266492
[math]f = z^2 \Rightarrow \tfrac{df}{dz} = 2z,\ \tfrac{df}{d\bar{z}} = 0[/math]
[math]g = \bar{z}^2 \Rightarrow \tfrac{dg}{dz} = 0,\ \tfrac{dg}{d\bar{z}} = 2\bar{z}[/math]
works nicely to me

Anonymous at Thu, 4 Jul 2024 05:36:21 UTC No. 16267253

>>16266492
>>16267243
maybe think of it like a variable change? Like [math]x = \tfrac{1}{2}(z + \bar{z}),\ y = \tfrac{1}{2i}(z - \bar{z})[/math]. This way you can do partial derivatives with respect to z and z-bar. Then, voila, wirtinger

Anonymous at Thu, 4 Jul 2024 06:32:29 UTC No. 16267307

>>16266482
Is your issue over [math]df = \sum_i \tfrac{\partial f}{\partial x_i} dx_i[/math]?
Isn't this just as bad as [math]\tfrac{d}{dx}(fg) = \tfrac{df}{dx}g + f\tfrac{dg}{dx}[/math]?

Anonymous at Thu, 4 Jul 2024 17:29:32 UTC No. 16267825

>>16266482
That is a partial derivative, correct?

Anonymous at Thu, 4 Jul 2024 17:30:53 UTC No. 16267828

>>16267243
>>16267253
>>16267307
Can you tell me what these mean? What version of Calculus it is? Is it Topology?

Anonymous at Fri, 5 Jul 2024 02:07:09 UTC No. 16268346

>>16267170
why does no one use Clifford algebra then?

Anonymous at Fri, 5 Jul 2024 03:22:34 UTC No. 16268395

>>16266482
I feel you, it is so cancerous people have the audacity of saying it somehow justifies infinitesimals and differentials. And yeah I don't understand why "geometers" wet themselves over this shit when it really it is much more algebraic, since it doesn't act locally but on a space. The worst is that in many courses all you see is just stupid manipulation of this notation while they appeal freely to the existence of smooth partitions of unity and solutions to odes like it is nothing but in reality this are the most important part of the proof. Stokes theorem is the most elegant shit only because the notation hides that in reality it is a a combinatorial application of the FTC. Where is the geometry here?

>>16267307
No because the derivative, as everyone knows, is a local object which is has a perfect geometric reasoing. The differential of a function assumes already that the function has regularity and shows nothing about the nature of the derivative. It only works because you already know calculus. It really is only useful if you are interested in topology and classification stuff but it is meaningless if you want to understand the meaning of shit like the derivative.

Anonymous at Fri, 5 Jul 2024 03:36:43 UTC No. 16268406

>>16267828
Normal calc and Multivariable calc. Like, intro.

Anonymous at Fri, 5 Jul 2024 08:31:35 UTC No. 16268700

>>16268406
Nah, the serious treatment of differential forms is taught at the graduate level when you're reading about manifold theory.

Anonymous at Fri, 5 Jul 2024 17:57:18 UTC No. 16269224

>>16268346
Clifford Algebra was mostly lost in a sea of papers up until around the 1980s when it David Hestenes found an popularized it.

It's gaining popularity, but keep in mind that most people who see this information just memorize the axioms and proofs to get through a class, and don't seriously make an attempt to streamline or learn the material.