@TanMath how you making out?

@TedShifrin hi

Hi @anakhro.

You might or might not cringe at my comments above.

I actually tend to call it the annihilator.

But nothing wrong with kernel.

@anakhro so sorry got busy with some other work. going to try and work this out right now...

@TanMath: If you're trying to develop some intuition for differential forms, my YouTube lectures might (or might not) be of some help.

@TedShifrin do you agree with my sentiment that visualizing differential forms is not as useful as learning how and where they are used?

I firmly believe that intuition comes with practice and examples.

There are lots of lectures, but that's the first. Make sure you find the replacement one at the end of the list — the last ten minutes of that video disappeared.

OK thanks I will look into it!

@TedShifrin intuition being a superset of visualization?

Yes.

I see.

@anakhro ok so I am used to finding the null space of matrices. So. would dx,dy,dz be basis vectors of a vector space?

And from that we could create a matrix representing the map $\alpha_p$?

I really do "visualize" vector fields, work, and flux ... I visualize differential forms that come up in the context of differential geometry — to me the connection forms are extremely geometric (twisting of the frame, etc.).

dx,dy,dz are the dual basis for T_pM

The differential form is a linear map on the tangent space, hence represents a row vector as a matrix.

@TedShifrin see that's nice... I read somewhere that a one-form is a bra vector. It's nice to see those kind of connections

@TedShifrin do you visualize differential forms as general objects? Like k-forms as per the generalization of that picture I shared above?

That's the Misner-Thorne-Wheeler picture. I've never taught it.

Again, I'd rather think about flux. of the vector field corresponding. But it's OK.

:( still not seeing though how the map can be represented as a matrix.

(my math's getting rusty :( )

@TanMath what is the definition of a 1-form that you are familiar with?

Actually that's another question I have for you guys. How do you guys refresh your math skills. For example, I haven't worked with linear algebra for a couple years now and I might need to refresh. How would I do that? do you guys just reread books and redo a bunch of problems

2. a p-form represents some "thing" you would integrate over, like a density

I am getting this all from Road to Reality btw

Have any of you read this book?

@TanMath if you already have done the material before and just need to brush up on stuff, I highly recommend doing an "exercises first" approach and then read back on chapters as you need to if you get stuck.

So just crack open a linear algebra book you have and do exercises from the start and move on as things come back to you slowly.

These two things you remember are not extremely helpful with regards to the exercise I gave you. I am not sure if that book of Penrose's gives you the actual definition of a manifold.

Do you know what a manifold is, @TanMath?

@anakhro I guess the general idea of it. The formal definition I don't remember. But there is an entire chapter dedicated to n-manifolds which is where the discussion on one-forms and p-forms started in the book.

@TanMath: My lectures cover multivariable analysis and linear algebra, including discussions of manifolds and different ways of thinking about them (not abstract ones, just ones in Euclidean space).

Yes, in the book he discussed how to construct manifolds from coordinate patches of n dimensions

@TedShifrin just Euclidean space? :(

I'm saying manifolds that live in Euclidean space ... no abstract charts.

That's the best way to learn until you're an expert, anyhow.

Unfortunately the book does not provide a precise definition of a manifold.

What book is this?

Road to Reality by Penrose

But it's not meant to be technical.

I used to own that, I think, but I got rid of most of my books.

@TedShifrin depends what you mean by technical. What do you mean by technical?

I mean written for people who know a lot of math and physics and using formulas ...

@TedShifrin It's got tons of formulas though.

Oh, OK. I don't remember.

Yeah 34 chapters of that good good math and physics. Starts out with geometry, even hyperbolic geometry, talks about numbers, complex numbers, calculus, complex calculus, surfaces, hypercomplex numbers, manifolds, group theory, fibre bundles. then starts going into physics: SR, GR, EM, QM, particle physics, QFT, cosmology, supersymmetry, and quantum gravity theories like string theory, loop quantum gravity, and twistor theory.

Ok so, if $A$ a subspace of a metric space has no isolated points then its diameter is equal to the diameter of its interior: we obviously have $\delta(\text{Int}(A)) \le \delta(A)$ and if $x, y$ are two points of $A$, since they're no isolated points there exists $x', y'$ where $x'$ is within $\epsilon$ of $x$ and $y'$ likewise.

By no means is it comprehensive though. for example it doesn't even have a formal definition of manifolds. I think Penrose it trying to focus on getting an intuition for these concepts, mainly a geometrical intuition. And I like it, but it starts getting a little confusing, given I am no math expert (have only gone till differential equations and some other math here and there)\

Is the proof correct?

@anakhro was there something you wanted to add regarding one-forms or you want me to continue with solving that problem you gave me?

Given a particular solution to an initial value problem, along with the initial condition, is there a way to reverse engineer the general solution?