12:00 AM
Hmm, yeah

That's what's going on.

Cool stuff.

LOL

That works for taking a 3-form to a 2-form, anyhow

i guess i'm just a moody kid after all
so much for being avant garde

12:01 AM
Something like that, grumpy kid.
@Semiclassic: $n$-volume and $(n-1)$-hypervolume :D

What about 2- to 1- and 1- to 0-form?

You can figure those out.

Compute area as length plus an additional vector.
(oriented)

Is it possible to do a discrete analogue of variational calculus by considering a 'curve' $\lambda$ made of countable points and then vary it to maximise/minimise an infinite sum $\sum_{\lambda} g({\lambda})$?

12:03 AM
I would highly doubt it, but who knows.

I think Mike made me figure out Cartan's magic formula by working the LHS and RHS out as derivations commuting with the boundary map in the chain complex. Doesn't make it unmagical to me, though.

The loose mental image I have is that a line is the intersection of two planes and a point is the intersection of three planes

That doesn't account for the bracket term, @Balarka.
I have no idea why you're doing that, @Semiclassic.

Well, i have in mind here a sequence of interior products taking dx dy dz -> dx dy -> dx

That doesn't really make sense. To get that you have to have specific vectors you're contracting against.
And it works left to right, actually, but up to sign ...
$\iota_{(a,b)}dx\wedge dy = -b\,dx + a\,dy$.

12:09 AM
In non-standard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal x {\displaystyle x} , the unique real x 0 {\displaystyle x_{0}} infinitely close to it, i.e. x − x 0 ...

So $\iota_{(a,b)} dx\wedge dy$ is the $1$-form whose value on $v$ is the signed area of the parallelogram spanned by $(a,b)$ and $v$.

I had in mind contracting with (0,0,1) and then with (0,1,0)

You actually have sign issues. You feed in at the left, not the right. But ok.

Yeah.
Should've done dy dz etc

Oh, the thirst answer on this link is what I just explained: math.stackexchange.com/questions/1997424/…
But with less motivation.
I think is a great answer to get a intuitive idea of differential.
Also, I think the ultralimits of Tao are a good approximation but I haven't read them deeply.

12:13 AM
That formula for $d\omega(X, Y)$ is really interesting. I have no idea what it geometrically means though.

And the notion in my head being, for instance, that the intersection of the planes y=0 and z=0 is a flow line in the (1,0,0) direction

Think about Stokes's Theorem on a tiny "curvilinear" parallelogram following the flows of $X$ and $Y$. (Of course, the bracket is there because that parallelogram might not close up.) @Balarka

But it's hard to say this right without a chalkboard

Just found something might be related to what I pondered: Slides on discrete lagrangian mechanics. The sample points are finite in number though.

Of course, I also am thinking in 3D, and that in itself is dangerous

12:15 AM

@Ted Ah!

(There is also a Hamiltonian version too)

So that's the infinitesimal Stokes' you were talking about. That's very fun.

Oh hell. I don't want power point.

I suspect what I'm doing only makes sense in 3D where there's a nice correspondence between vector fields and 2-forms

12:17 AM
No, I don't think that's relevant to this, @Semiclassic.

OK, I have to eat something and then go out for the evening. Bye, all. Good un-sleep, Balarka. Glad you are slightly less grumpy :D

Is Noah Schweber here?

Enjoy!

Never here, @law-of-fives.

12:18 AM
ok

Wow, there is so much information on math stackexchange about the differentials.
Why does it confuse a lot of people?