TanMath

If you know of any, otherwise it's fine if you don't...

Well just an intro to the notation would be nice, because it seems that Penrose will demonstrate its use throughout the book

Do you know of any good tutorials on this notation? because Penrose just quickly skimmed over it at the end of a chapter in the book

Also, it doesn't feel very useful (at least right now) like bra-ket notafion, which is immediately very helpful for understanding linear algebra in the context of QM

So it does make things clearer? Because looking at it now, it seems pretty confusing

Like it helps people obtain new results in math and physics

Sorry I am asking such random questions, just trying to go through this book I am reading...

Is Penrose diagrammatic notation something that's actually useful in math and physics?

Thanks... Seeing that just now on the wikipedia page of groups

@TedShifrin ah okay the symmetrics lead to specialized types of groups? But in general a group can be any sort of defined set with a defined set of operations?

What is a group? is it like a set of elements with different symmetries?

@anakhro ah i see that... It seems like a pretty good geometrical picture of one-forms though

@Semiclassical This may be true for math or physics students who might learn these things in their curriculum. But I am reading road to reality on my free time, to have a better grasp of math and physics. Indeed, there are some concepts I need to look up here and there. But there are definitely details I don't understand, and even Penrose says in the Preface that's totally OK.

@anakhro yah

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

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)\

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.

@TedShifrin It's got tons of formulas though.

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

It's actually a fun book to read ;)

Road to Reality by Penrose

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

@TedShifrin just Euclidean space? :(

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

@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.

Have any of you read this book?

I am getting this all from Road to Reality btw

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

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

(my math's getting rusty :( )

:( still not seeing though how the map can be represented 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

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

@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?

OK thanks I will look into it!

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

Well I am not used to maps, except as the concept that almost everything is a map lol

Oh wait by kernel you mean a null space?

@anakhro sorry I am not following along. I don't know what you mean by kernel.

@anakhro yes this is what I am thinking of. Where is this from BTW?

I have heard that for a one-form, since the scalar product of a one-form with a vector is a scalar (that's the definition after all), the one-form can be represented by a plate that the vector crosses to result in a scalar... is this similar to the interpretation you are describing here?

What do you mean by the kernel of a one-form?

Their geometrical intuition. like what they mean geometrically/physically. I can "see" a vector and a vector field, but it is harder to do so with a one-form and p-forms

yes

Also, any resources on Clifford and Grasmann algebras? Those sections in Road to Reality were a little confusing

I wonder if you guys have any resources for better understanding one-forms and p-forms? I guess I have some understanding of it, but I would like to improve my geometrical intuition of it.

what do people think of the latest rep change?