/sci/ - Science & Math » Thread #13431344

32KiB, 220x253, 32CB1786-2ACD-4996-AF76-1B39E1906222.png

View Same Google iqdb SauceNAO

Cross product is a thing of the past.

Anonymous Sat 24 Jul 19:16:41 2021 No.13431344 View Reply Original Report

Quoted By: >>13431377 >>13431394 >>13431398 >>13431446 >>13431453 >>13431535 >>13432592 >>13432993 >>13438690 >>13442597 >>13444756 >>13445537

https://www.youtube.com/watch?v=60z_hpEAtD8

Fuck cross product, geometric product is what we should've been learning from the beginning.

Anonymous

Anonymous Sat 24 Jul 2021 19:24:02 No.13431377 Report

Quoted By: >>13431398

>>13431344
I've been adapting my tutoring materials to geometric algebra for a while now, first product is free.

Anonymous

Anonymous Sat 24 Jul 2021 19:30:21 No.13431394 Report

Quoted By: >>13431661

>>13431344
I've been using wedges since calc 3 because they literally do the jacobian for you.

Anonymous

Anonymous Sat 24 Jul 2021 19:31:40 No.13431398 Report

Quoted By:

>>13431344
>>13431377

This sounds like a cult like those category theorists.

Anonymous

Anonymous Sat 24 Jul 2021 19:47:25 No.13431446 Report

Quoted By:

>>13431344
How do you use this exactly?

Anonymous

Anonymous Sat 24 Jul 2021 19:49:18 No.13431453 Report

Quoted By:

>>13431344
I thought this was about exterior algebras

Anonymous

Anonymous Sat 24 Jul 2021 20:16:18 No.13431535 Report

Quoted By: >>13431590 >>13431717 >>13432333 >>13432550

>>13431344
interesting video.
unfortunately i think to appreciate the generality you really need to know about all the examples.
moreover, it's important to get comfortable with not always having a geometric interpretation.
also, reducing a set of equations down to one (he does this to maxwell's equations) really is just hiding things behind layers of notation as he points out.
useful stuff, i like it. but also i think it doesn't make anything obsolete

Anonymous

Anonymous Sat 24 Jul 2021 20:37:08 No.13431590 Report

Quoted By:

>>13431535
What the fuck

He stole this from me

Thia
S

Anonymous

Anonymous Sat 24 Jul 2021 21:08:16 No.13431661 Report

Quoted By: >>13431669 >>13431676 >>13432973 >>13432975

>>13431394
>jacobian
The Jacobian is a matrix. The Jacobian does not have to be square. If it is a "square" system, then $\wedge_{k=1}^n \omega_n = \det (A) \wedge_{k=1}^n _alpha_n$ if $\alpha_n$ is a vector and $(\omega_k) = (\alpha_k) A$ defines a set of $n$ vectors. If your system is not square, i.e. $n$ vectors is a space of dimension $m< n$ , you have no determinant since everything is identically zero. If $m>n$ , then you have a set of $\binom{m, n}$ coefficients that could be interpreted as sub-determinants (should be checked).

Here's an example of a non-square Jacobian. The Jacobian of the function $f:\mathbb{R}^2 \to\mathbb{R}^3$ defined by $f(x,y) = (\cos(u), \sin(u), v )$ has the determinant
$Df(x,y) = \begin{bmatrix} <br /> -\sin(u) & 0 \\<br /> \cos(u) & 0 \\<br /> 0 & 1 \\<br /> \end{bmatrix}$
Is not square

Anonymous

Anonymous Sat 24 Jul 2021 21:11:35 No.13431669 Report

Quoted By: >>13432973 >>13432975

>>13431661
> $\alpha_n$ is a vector
should be
$\alpha_k; k=1,..., n$ is a basis.
Don't know why the other code isn't working
$\wedge_{k=1}^n \omega_n = \det (A) \wedge_{k=1}^n _alpha_n$
$\binom{m, n}$

Anonymous

Anonymous Sat 24 Jul 2021 21:13:47 No.13431676 Report

Quoted By: >>13431700

>>13431661
>coefficients that could be interpreted as sub-determinants
The wedge gets those right too.

Anonymous

Anonymous Sat 24 Jul 2021 21:20:27 No.13431700 Report

Quoted By:

>>13431676
If you are integrating f(x,y,z) dxdydz, first write it as
f(x,y,z) dx^dy^dz.
then change variables x = x(r,s,t), y = y(r,s,t), z = z(r,s,t).
dx = (xr)dr + (xs)ds + (xt)dt
dy = (yr)dr + (ys)ds + (yt)dt
dz = (zr)dr + (zs)ds + (zt)dt
performing the wedge product (ie distributing it) gives the correct integrating factor *dr^ds^dt

Anonymous

View Same Google iqdb SauceNAO charge-density.png, 105KiB, 641x1083

Anonymous Sat 24 Jul 2021 21:27:33 No.13431717 Report

Quoted By: >>13437961

>>13431535
I think it may be worth it. The idea of considering charge density as a current moving through time instead of space is quite thought provoking

Anonymous

Anonymous Sun 25 Jul 2021 01:55:34 No.13432333 Report

Quoted By: >>13432668

>>13431535
While true, the meanings behind the equations are not obfuscated. I'd say it more clear since the everything is compacted down to one equation, and operations follow a common pattern.

Anonymous

Anonymous Sun 25 Jul 2021 03:52:41 No.13432550 Report

Quoted By:

>>13431535
>unfortunately i think to appreciate the generality you really need to know about all the examples.
I don't think so, well, at least in the most basic parts. I'm not a physicist and for me torque never made any fucking sense, but seeing it as a bivector has finally clicked in my mind.

Anonymous

Anonymous Sun 25 Jul 2021 04:10:17 No.13432592 Report

Quoted By:

>>13431344
physics.... yawn

Anonymous

View Same Google iqdb SauceNAO torque-bivector.png, 66KiB, 1134x640

Anonymous Sun 25 Jul 2021 04:47:12 No.13432668 Report

Quoted By:

>>13432333
checked, that I found that part beautiful too

Anonymous

Anonymous Sun 25 Jul 2021 07:18:26 No.13432973 Report

Quoted By:

>>13431661
>>13431669
What the fuck does
> $\wedge_{k=1}^n\omega_n=\det(A)\wedge_{k=1}^{n_{\alpha_n}}$
> $\binom{m}{n}$
mean? I suck at math...
>Jacobian
God, I don't even know where to start to be honest.

Anonymous

Anonymous Sun 25 Jul 2021 07:20:17 No.13432975 Report

Quoted By: >>13432978 >>13433003

>>13431661
>>13431669
What the fuck does
> $\wedge_{k=1}^n\omega_n=\det(A)\wedge_{k=1}^{n_{\alpha_n}}$

> $\binom{m}{n}$
mean? I suck at math...
>Jacobian
God, I don't even know where to start to be honest.

Anonymous

Anonymous Sun 25 Jul 2021 07:21:31 No.13432978 Report

Quoted By: >>13433003

>>13432975
Nevermind, I actually know what the second thing is but what is the first thing?

Anonymous

Anonymous Sun 25 Jul 2021 07:30:18 No.13432993 Report

Quoted By: >>13433034 >>13439688

>>13431344
Sorry, but GAs fucking suck, the notation is very cumbursome.

The Exterior Calculus is going to be taught in the future. It's leaner, more intuitive, more powerful and connects its notation more closely to differential geometry and other fields of math.

The GA shills are literally what's holding universities back ftom teaching exterior products instead of multivariable calc. All because some fag wants to garner an ungodly number of citations.

Anonymous

Anonymous Sun 25 Jul 2021 07:36:44 No.13433003 Report

Quoted By: >>13433053

Exterior products are nice and all but if you really want the children to understand angular velocities, you should teach them about lie groups and lie algebras, particularly $\mathfrak{so}(3)$ .

>>13432975
>>13432978
>I suck at math...
You don't need to worry since this is not part of the standard curricula even for engineers. If you want to know what they are, look up wedge products.

Anonymous

Anonymous Sun 25 Jul 2021 08:02:52 No.13433034 Report

Quoted By: >>13433043

>>13432993
aren't those two the same thing?

Anonymous

Anonymous Sun 25 Jul 2021 08:09:35 No.13433043 Report

Quoted By: >>13433263 >>13438123

>>13433034
No, not at all, they come from different disciplines and have different names and more importantly have different properties. The geometric product is the only contribution of GA that is the claim to fame of the field.

The problem with GA objects is that to define the geometric product on it, you lose basically every powerfull theorem from differential geometry. I don't expect physicists or even mathematicians to understand, but as an engineer who has contributed to development of both GAs and discrete exterior calculus (DEC) libraries I can promise you that the former has no future compared to the latter. Due to implementation detail difference, they are not and never can be the same thing. DEC has won already, it has been picked by Nvidia and Intel, in the future hardware optimised for bilinear operations will use DEC libraries.

Anonymous

Anonymous Sun 25 Jul 2021 08:13:55 No.13433053 Report

Quoted By: >>13433217

>>13433003
>if you really want the children to understand angular velocities, you should teach them about lie groups and lie algebras, particularly so(3).
Interesting. Is there a good textbooks that you would recommend? Honestly I've only seen Lie algebras is Mathematical Physics books that will certainly fly over the heads of second years.

Anonymous

Anonymous Sun 25 Jul 2021 08:58:54 No.13433121 Report

Quoted By:

Imagine getting filtered by cross products then thinking learning geometric algebra would be easy

Anonymous

Anonymous Sun 25 Jul 2021 10:01:30 No.13433217 Report

Quoted By: >>13433269

>>13433053
I first saw this in Spivak's physics for mathematicians, but it'll probably be covered in any sufficiently advanced classical mechanics or robotics textbooks too. Roughly speaking, the configuration space of a rigid body spinning about a fixed point in the body can be modeled as the group $G = SO(3)$ and choosing a metric on TG is basically equivalent to defining the moment of inertia of the body (once you have the metric, you can define the kinetic energy and follow the usual Lagrangian mechanics approach). Now you can formulate all of this using only the language of rotation matrices (SO(3)) without ever mentioning general Lie groups at all (which is what Spivak's book does, more or less) so I think it could probably be explained to students who are familiar with some linear algebra and multivariable calculus, but I haven't really thought through this in detail.

Anonymous

Anonymous Sun 25 Jul 2021 10:31:51 No.13433263 Report

Quoted By: >>13433274

>>13433043
forgive me my ignorance but how can those two be compared at all?
One of them is analysis(calculus), the other is algebra. I was under the impression you were comparing Geometric Algebra and Exterior Algebra

Anonymous

Anonymous Sun 25 Jul 2021 10:35:28 No.13433269 Report

Quoted By:

>>13433217
Wow, thank you Anon this is unironically a very good practical example that is completely absent from texts like Szekeres' Mathematical Physics etc.

I will pick up Spivak and see if we can use it in our undergrad course.

Anonymous

Anonymous Sun 25 Jul 2021 10:37:16 No.13433274 Report

Quoted By:

>>13433263
Apologies. I meant Geometric Calculus not Geometric Algebra (which is what linear algebra is to multivariable calculus).

Anonymous

Anonymous Sun 25 Jul 2021 23:58:17 No.13435939 Report

Quoted By:

Why not simply the tensor algebra? Then you can define the wedge product as the commutator and the inner product as the trace of the product.

Anonymous

Anonymous Mon 26 Jul 2021 13:58:09 No.13437961 Report

Quoted By: >>13440126

>>13431717
time machine? tictacs?

Anonymous

Anonymous Mon 26 Jul 2021 14:57:28 No.13438123 Report

Quoted By: >>13438666 >>13439339

>>13433043
it's a generalization, you don't lose shit
all generalizations subsume previous research

like using complex numbers includes all results on the reals

Anonymous

Anonymous Mon 26 Jul 2021 18:16:20 No.13438666 Report

Quoted By:

>>13438123
>like using complex numbers includes all results on the reals
real and complex analysis are different enough that you don't get real analysis for free by knowing complex analysis

Anonymous

Anonymous Mon 26 Jul 2021 18:24:14 No.13438690 Report

Quoted By: >>13438739 >>13438943

>>13431344
how the fuk do you add a vector to a scalar, pseud?

Anonymous

Anonymous Mon 26 Jul 2021 18:36:39 No.13438739 Report

Quoted By:

>>13438690
Like how you add a real number to an imaginary number to get a complex number you brainlet.

Anonymous

Anonymous Mon 26 Jul 2021 19:43:35 No.13438943 Report

Quoted By:

>>13438690
That's what they said about schoolteacher Grassmann
Also wasn't there a story about Feynman "cheating calculations" using Clifford algebra, which none of the other physicists knew?

Anonymous

Anonymous Mon 26 Jul 2021 22:50:31 No.13439339 Report

Quoted By:

>>13438123
>like using complex numbers includes all results on the reals
Not really. For example, complex differentiability turns out to be a much stronger condition than real differentiability. Analyticity, holomorphicity, and conformality turn out to be equivalent in the complex numbers, but not in other settings.

Anonymous

Anonymous Tue 27 Jul 2021 01:23:52 No.13439688 Report

Quoted By:

>>13432993
Geometric algebra is not taught anyway. If you're going to complain about someone, complain about the one's who insist on teaching vector calculus. At least GA would be a step in the right direction.

Anonymous

View Same Google iqdb SauceNAO magnetic-vector-potential.jpg, 42KiB, 891x508

Anonymous Tue 27 Jul 2021 04:42:52 No.13440126 Report

Quoted By:

>>13437961
That's the target

Anonymous

Anonymous Tue 27 Jul 2021 21:43:58 No.13442597 Report

Quoted By:

>>13431344
Any actual introductions with more examples and clear definitions on this subject?

Anonymous

Anonymous Wed 28 Jul 2021 14:40:29 No.13444756 Report

Quoted By:

>>13431344
Thanks, fascinating video.

Anonymous

View Same Google iqdb SauceNAO dimensions.png, 6KiB, 681x252

Anonymous Wed 28 Jul 2021 18:44:02 No.13445537 Report

Quoted By: >>13446519 >>13449069

>>13431344
I have been using this shit for 3d graphics for a few years, it makes life so much easier. I can even work out all sorts of transformations on paper by hand. No stupid matrices

Anonymous

Anonymous Wed 28 Jul 2021 23:28:35 No.13446519 Report

Quoted By:

>>13445537
Have you like written a library for it?

Anonymous

Anonymous Thu 29 Jul 2021 16:47:14 No.13449069 Report

Quoted By:

>>13445537
bullshit

Capcode	All Only User Posts Only Moderator Posts Only Admin Posts Only Developer Posts
Show Posts	All Only With Images Only Without Images
Deleted Posts	All Only Deleted Posts Only Non-Deleted Posts
Ghost Posts	All Only Ghost Posts Only Non-Ghost Posts
Post Type	All Only Sticky Threads Only Opening Posts Only Reply Posts
Results	All Grouped By Threads
Order	Latest Posts First Oldest Posts First

Your latest searches