Ted quick question about generalizing the result of integrating an odd function over a symmetric rectangle. I take it for the claim to stand in a general region, we would need the region to be symmetric in some way? Is that what you meant by generalizing it?

Awesome

thank you

Then write $f(G,C)$, where $G=(V,E)$. You can think of functions whose domain is $\Bbb R^2\times\Bbb R$ rather than $\Bbb R^3$.

I don’t remember the context, DC. You can certainly generalize to regions that are symmetric with respect to the appropriate hyperplane or ….

In your question, all the hyeprplanes were set at the origin

Could be lower-dimensional, too.

Ok then.

Like $f(x,y,z)=f(-x,-y,z)$

yea. that's exactly the type of function you defined

That’s symmetric about a line, rather than plane. Right?

Ah...I see what you're getting at. I have to ponder on it a lil bit more then

I need more help. How to define a "form" of something. I want to say we can get \hat{DG} of DG given a DG and X

I want to say hat{DG} of DG is a function that takes a DG and something else and returns ...

A general relativity book I'm reading states that given a coordinate chart $\varphi=(x^1,...x^n)$, since the object $dx^1\wedge...\wedge dx^n$ under a coordinate transformation to $(y^1...y^n)$, we have $\mathrm{det}J dy^1\wedge...\wedge dy^n$ (where $J$ is the Jacobian matrix of the coordinate transformation), it is not a differential n-form. Instead it is a tensor density. Isn't that completely wrong?

Anytime we pull back a $n$-form between $n$-manifolds we get a jacobian determinant

Also, even thinking in terms of "transformation laws", that is the transformation law of a $n$-form, just written more compactly with the determinant (that contains the various $\partial x^k/\partial y^h$). A tensor density would require both the determinant and the $\partial x^k/\partial y^h$. I think my source is confused between the tensor densities and form invariance

Count on GR people to mess up math. Are you sure they didn’t put absolute value of the det? That would make it a density.

Even then, to me they are just rewriting how the n-form transforms. I'll show you asap that page

I don’t follow your last paragraph at all.

You are right, that's why I'm posting the paragraph of the book

OK

I think this is just a huge - wrong - handwave to say that $dx^1\wedge...\wedge dx^n\neq dy^1\wedge...\wedge dy^n$ in general

I would not trust these authors.

They are not mentioning orientation issues, which is the point of $n$-form versus $n$-density.

What’s more, using absolute value sign for determinant makes it more confusing.

I'm giving the differential geometry preface a quick read just to make sure I know their notation but seeing diff geometry in a GR book after having studying rigorous diff geo is traumatic

Is this a respected published book or someone’s typeset notes?

By the way, I checked the notation and they mean the determinant, not its absolute value. With a more acceptable notation, they are saying that $dx^1\wedge...\wedge dx^n=\left(\mathrm{det}\frac{\partial x}{\partial y}\right)dy^1\wedge...\wedge dy^n$ is a density behaviour, which is bullshit

I know they meant det. I said so.

@TedShifrin it is a respected book indeed, it's Carroll's book on GR. I wouldn't doubt the GR part but the diff geo part looks kinda sloppy

Well, since GR depends on DG as a foundation, I would doubt it.

I'll check for possible errata corrige tomorrow. In case I don't find any, do you think the author should be mailed to correct this in future editions?

There are going to be compounding issues if they misunderstand easy basics.

I see. I'm kinda regretting purchasing the book now but whatever. Good night (1pm here!) and thank you

I just looked this up. Originally published 20 years ago.

Night.