Mathematics - 2021-11-01 (page 4 of 4)

« first day (4107 days earlier) ← previous day next day → last day (1211 days later) »

Semiclassical

23:05

@TedShifrin by contrast, I feel like physicists just deal with differentials (in the intuitive sense) and don’t think about them as forms

In keeping with that, a “vector field” for a physicist is usually just a vector-valued function

Albeit one which behaves in certain ways under rotations

leslie townes

that is reflective of my limited experience. although every department has a number of the good sort of physicist who at least speaks the language of forms, even if they don't think in it.

Ted Shifrin

I disagree with that.

Even my colleagues at UGA who’d been math majors shunned forms like the plague.

leslie townes

the ones i know would say it's more than a box of numbers that transforms a certain way, but they wouldn't use forms language properly either.

they talk about things that have meaning and things that don't. it's all metaphysical.

Ted Shifrin

No, leslie, you’re thinking tensors. Different world.

Semiclassical

If the measuring stick of forms is “eats tangent vectors and spits out numbers” then physicists don’t rate highly on that

Ted Shifrin

23:11

Relativists have to use those, and the moment of inertia tensor is in mechanics. But different world.

leslie townes

i was thinking stuff like with maxwell's equations. what's that?

Semiclassical

By contrast, the notion that differential forms are gadgets which transform under rotations differently than vector fields wouldn’t be so strange to a physicist

But that doesn’t really exhaust the concept of a differential form

@leslietownes I have seen Maxwell’s equations in the language of forms

leslie townes

i remember wanting to tell the outside member on my thesis committee (physicist) that he wasn't actually supposed to ask me real questions about my thesis. he was just supposed to sign on the line like all the other outside members do.

unfortunately he took his job seriously.

Ted Shifrin

I have a section in my book on Maxwell with forms. Math people do that, not physicists.

Semiclassical

And it does create some interesting effects in the formalism. The electric field turns into a certain one-form iirc, while the magnetic field turns into a two-form

dc3rd

23:16

Howdy Math group...everybody toiling along?

leslie townes

FORMalism. get it ???

Semiclassical

Which is interesting for classical electrodynamics, insofar as it puts the electric and magnetic fields on distinctly different footing

leslie townes

ha, ha

Ted Shifrin

Plenty of Hodge star operators running around.

Semiclassical

But there’s also a formulation in terms of 3+1 space time, in which case the electric and magnetic fields are both two-forms

dc3rd

23:18

.how nice just the guys I wanted to ask a question to....or rather verify my understanding of the big picture.

Semiclassical

And then something something field strength tensor

Slate

...I'd be really curious to read that, @TedShifrin.

Semiclassical

Most physicists just deal with that via index notation shenanigans tho

Ted Shifrin

Yup.

Semiclassical

“Shut up and use index notation”

leslie townes

23:32

i own two physics books. i've never read them but they are littered with index notation.

Semiclassical

index manipulations can start to feel like gymnastics

though a lot of the time it amounts to doing the BAC-CAB rule for the vector triple product

just written in index form

dc3rd

It had to do with the "bigger picture" in regards to the question you guys helped me with last night. I want to make sure I'm putting all the moving parts into the right places... (I really need to stop with this business "buzzword" phrasing that I use)

So the the machinations in the mind of dc3rd:

"We have this object (in this case it was in $\mathbb{R}^4$). We don't know anything about this object except for the expression that was given. One way of us trying to build an understanding of this object is to perhaps attempt to get "linear approximations" of the object around points. So we take the derivative, which tells us that we can write one of the coordinates that describe our object
as a function (graph) of the other coordinates (in the case of the exercise it was in terms of two coordinates). This coordinate that could be written a…

Ted Shifrin

Why are you saying $\Bbb R^4$?

In general, understanding from an equation what an object looks like (even how many connected components it has) is virtually impossible.

Semiclassical

it's definitely the exception rather than the rule

it is an interesting question, as i think of it: obviously, humans can't visualize in four dimensions. (closest is stuff like looking at 3D objects evolving in time)

Ted Shifrin

I don’t know where 4D came from here.

Semiclassical

23:45

the mention of RR^4. i'm getting philosophical

dc3rd

I was saying $\mathbb{R}^4$ because for the "object" in general took $(x,y,z)$ as its entries and produced a fourth value so I was picturing the same thing you would picture if we had a function $z = f(x,y)$ and we were to draw it, it would be something in $\mathbb{R}^3$....so for example the cylinder

Ted Shifrin

No one is talking about the graph. Don’t confuse yourself yet again.

Semiclassical

what i'm getting to, obliquely: what would it even mean for a (non-human) mind to visualize 4D? i can't quite figure out how to even pose the question

dc3rd

hmmm....so I was trying to get a "visual" of what was going on with the object....So what I just said right above this corresponds to the idea of the "graph", but the function itself doesn't necessarily have a "graph" so it can't be visualized as such...

Semiclassical

do you have a specific example in mind? it's hard to be concrete without that

Ted Shifrin

23:48

Now you’re saying nonsense.

Semiclassical

i'd ask who that's directed to, but probably "both"

Ted Shifrin

He’s thinking of our long discussion last night.

Semiclassical

leslie townes

a fairly common abstract theme is to try to understand something by understanding lower-dimensional pieces of it. this could be as simple as given a set characterized by an equation and actually finding one or more tuples that satisfy the equation. often there is no 'galaxy brain' moment.

oh, i missed last night. feel free to ignore me. usually the safest course of action.

Semiclassical

was last night the example of the twisted cubic?

dc3rd

23:51

Ok, let me bring together in an example. So we are given $f(x,y,z) = x^2+y^3+xyz^3$. We do all the differential stuff and it turns out it is a smooth surface.

Ted Shifrin

The tangent developable thereof, yes.

No. What is the surface?

fin

Im back

Semiclassical

a fun surface: $f(x,y,z)=1+2xyz-x^2-y^2-z^2=0$

dc3rd

So I know we can't "visualize" what happens in the 4th dimension, but I was trying to draw a parallel to what we do in 3 dimensions where we would use $x,y$ and then the value of $z = f(x,y)$ would give us a 3 dimensional object.

Semiclassical

so something like $w=f(x,y,z)$?

dc3rd

23:56

yes

Ted Shifrin

We were not talking about graphs. Not recently at all.

dc3rd

maybe another way I can explain what I'm thinking is, I'm trying to visualize things the way that it is done in Flatland

Semiclassical

to get at what Ted's saying: the set of points $(x,y,z,w)$ such that $w=f(x,y,z)$ is by definition the graph of $f$

dc3rd