« first day (4914 days earlier)      last day (30 days later) » 

12:16 AM
okay this is my fast and loose summary of the discussion on differential forms over a differentiable manifold :P
 
 
1 hour later…
1:27 AM
@ACuriousMind what does this mean
@SillyGoose u eldritch being
 
2:17 AM
u shellshermichael @Relativisticcucumber
 
3:02 AM
@SillyGoose you said you define the exterior derivative but you didn't
Anyhow what's the name of the font
 
@lucabtz oh yes i wrote "we define the usual..." so i didn't have to write down the definition XD but i should add that in...
let me check about hte font
im not sure it might be called "libertine"
 
@SillyGoose thanks
 
this latex template also uses this package which seems to affect the font: \usepackage{mathptmx} % d. ...in a simple roman face except where indicated below (§3).
 
 
2 hours later…
5:07 AM
@SillyGoose you can always go back to old Newtonian mechanics with every single particle in physical space instead of the combined configuration space~
 
meow
 
@SillyGoose KathyLovesPhysics covers the original papers of E&M. Extremely amazing stuff.
@SillyGoose M I A O ~ ~
@SillyGoose It cannot be, because the completion of putting together Maxwell's equations were all done without the conception of SR nor QM. Of course, if you want a modern account that emphasises how SR and QM are actually really needed for Maxwell's equations to be as they are, you might find one, but it is not how we got here.
Also, ACM's reply to you there is undoubtedly correct, but he omitted an important aspect, probably because everybody talks about this: The spin-statistics theorem says that the spin one of photons makes it so that photons obey Bose-Einstein statistics, and this amplifies the collective wave aspects, so that the classical wave aspects are salient. Contrasted with electrons obeying Fermi-Dirac statistics, which amplifies individual particle aspects.
@SillyGoose burn this. Either define the 1-form with a minus sign, or define the 2-form with the opposite sign. Nobody should be taking the negative of a exterior derivative as a convention.
 
I have a probably really simple question about an example in Lancaster and Blundell's QFT book:
In 1.37 how does the derivative of $\tfrac12 (\partial_\mu\phi)^2$ end up as $\partial^\mu\phi$ rather than $\partial_\mu\phi$?
Presumably it is related to the fact that $(\partial_\mu\phi)^2 = (\partial_\mu\phi)(\partial^\mu\phi)$?
 
5:25 AM
Is it possible that I have finally found a moment where I could help John Rennie?
 
@SillyGoose horrifying mistake, but it is also the kind of mistake that would be completely prevented if you had concrete applications of all the various entities involved, instead of an abstract-only learning. It is not just the metric that is a symmetric tensor. There are plenty of others in physics.
@JohnRennie yes
 
@JohnRennie i think if you write it out as you did and then use the metric to lower the index of the second term and then differentiate using product rule you will get your wanted result
 
I think $$\frac{\partial}{(\partial_{\mu}\phi)}\frac{1}{2}(\partial_\mu\phi)^2 = \frac{2}{2}\frac{\partial}{(\partial_{\mu}\phi)}(\partial_\mu\phi)(\partial^{u}\phi)$$
 
In particular, it is that $\frac12(\partial_\mu\phi)^2=\frac12\partial_\alpha\phi g^{\alpha\beta}\partial_\beta\phi$ so that the derivative denominator that is $\partial\partial_\mu\phi$ with the two partials, converts first just one, and then the other, into a kronecker delta. Then the two parts are the same thing, and cancel away with the half.
@Obliv why do you have a dangling $\partial^\mu$ at the end?
 
woops
 
5:30 AM
@Obliv the new edit is straight up false...
 
welp too late to edit it now
I shoulda put a bracket around those two terms
it's meant to be partial of those last 2 terms which should just be $\partial^u\phi$
 
or whatever that var. is in the superscript
 
writing index notation in latex is tiresome XD
 
@SillyGoose This package defines Adobe Times Roman (or equivalent) as default text font, and provides maths support using glyphs from the Symbol, Chancery and Computer Modern fonts together with letters, etc., from Times Roman. It supersedes both the original times and the mathptm packages.
 
5:32 AM
the bottom half of the page is the correct calculation (im pretty sure)
i obtain $2$ times the correct result because i left out the factor of $\frac{1}{2}$
 
@JohnRennie I have no idea what's going on physically/mathematically but it seems to me if you plug in that identity and then take the derivative, the term $\partial_{\mu}\phi$ vanishes and we have $(\partial^u\phi)$ kinda like $\frac{\partial}{\partial x}xy = y$ where $x = \partial_{\mu}\phi$ and $y = \partial^u\phi$
 
@naturallyInconsistent oh
 
@SillyGoose it is. It skipped the important step, though
@Obliv no, this is not what is happening
 
Of course, yes, silly me :-)
Thanks everyone :-)
 
5:34 AM
I mean that's literally what's happening @naturallyInconsistent
 
It is not a usual day that we get to help Rennie
@Obliv and I am telling you that you are wrong on that
 
In what way?
 
@Obliv $(\partial^\mu \phi)$ is not an independent variable
from $\partial_\mu \phi$
 
@Obliv The correct reason is already written by the wacky fowl
If you want it in more detail and LaTeXed goodness:
 
if you wanted to do the calculation systematically, you should lower all indices via the metric tensor and then apply usual calculus to take derivatives
 
5:36 AM
where did the $\eta$ come from
what da dog doin
 
Minkowski metric
It's being used to lower the index.
 
I like my explanation better because I can understand it, even though it's wrong :D
 
$$\begin{align}\frac{\partial\ \ }{\partial\partial_\mu\phi}\frac12(\partial_\nu\phi)^2&=\frac12\frac{\partial\ \ }{\partial\partial_\mu\phi}\partial_\alpha\phi\,g^{\alpha\beta}\,\partial_\beta\phi\\&=\frac12\left[\delta_\alpha^\mu\,g^{\alpha\beta}\partial_\beta\phi+\partial_\alpha\phi\,g^{\alpha\beta}\delta_\beta^\mu\right]\\&=\partial^\mu\phi\end {align}$$
@Obliv that's a paddling
 
so what do the subscript/superscripts represent
 
@Obliv contravariant v.s. covariant indices
 
5:42 AM
can I learn this power
 
@Obliv you should
 
I thought my intro abstract algebra course would touch on concepts like vector spaces,modules, etc
i guess technically we did touch on vector spaces via fields
but not explicitly
I'm just gonna learn on my own what manifolds are and tensors etc
all the important stuff in physics
didn't even know this was a subject of its own
meow
 
@Obliv technically, none of it is of any importance. Plenty of physicists get by with none of these. But it is very helpful to know some of these maths
M I A O ~ ~
 
I guess you're right, but it feels like I've come this far, so might as well put in the extra effort to learn a few more abstract concepts
plus then I can decipher what is going on with these scribbles in the h-bar
 
yes, these are very helpful stuff to learn, but unfortunately many coverage are just bad
 
5:49 AM
it looks like truly nonsense from my PoV :P
@naturallyInconsistent do u know of a good resource for deriving the ensembles in stat mech
Schroeder served me well in like the first couple chapters but I'm starting to see the shortcomings
it's probably still my best resource cuz the curriculum follows it pretty closely
 
@Obliv You should try Ian Ford Statistical Physics, an Entropic Approach. If you want a lot of maths, do Callen.
Callen does a lot of the physical arguments properly.
 
Okay, will check those out. Thanks :)
 
@Relativisticcucumber incomprehensible utterances are a hallmark of eldritch beings a la Lovecraft - once you understand them, you typically go insane, i. e. become a physicist
 
oh no the mods are no longer asleep. shoos the animals out of the bar
 
M I A O !!!
 
6:00 AM
We have geese just chillin on campus snacking on the grass and they're so docile/used to students just walking near them.. but I have to remind myself that they're actually just cobra chickens..
no offense @SillyGoose
 
I wonder if I should admit we had roast goose for the big Easter meal.
It was very tasty!
 
6:14 AM
i will admit that i think duck is one of the greatest foods ever
@Obliv omg i wish we had geese
we have squirrels and lizards
 
6:32 AM
@ACuriousMind lol
my supervisor believes that nobody really understands physics and that people just say random things that they think are related to what is being discussed
i think thats a him problem
i have begun to dig too deep into his endeavors. i have found that 30% of his citations are from himself. on average, 70% of the citations of his papers are by him
or "on median" im not sure which i should say. selecting papers at random, examining who they have been cited by, 70% of it is him is what i mean
 
zoinks
 
@Relativisticcucumber omg that is something that I would consider for reporting to the university. Can you, like, pivot to another supervisor or something? Very very scary
 
im leaving in august anyways but yeah i didnt realize how bad this guy was until recently
some events have made me think quite negatively of him and i began to realize more and more bad things once i lost my initial respect for him
sadly i have come to realize many figures in science are corrupt :( and it seems that trying to do anything ab this just gets u blacklisted :(
 
6:50 AM
Sigh
That is also why niche parts of science deserve scrutiny. It is much more difficult to make shit up in hotly contested experimental areas, whereas if you write shit in a niche area that nobody really cares about, you can hide quite well
 
yeah well this guy is quantum info but hes not that big. it just bothers me bc he refers to himself as someone who is big in the field and its just annoying bc i wanna be like bro i know u r just a citation machine with urself who r u fooling
but it does make me sad that science is corrupt. but its just smth to learn to navigate i guess : ////
 
@Relativisticcucumber MIT be like: 0.0
 
but ill just learn as much as i can and then onto bigger and better things in august !!!
 
@Relativisticcucumber omg so many horror stories of QC. Very very annoying
 
@SillyGoose MIT be like "lets name this new building after a pedophile!"
lets visit sex offenders in jail!
still better than stanford
there the offenders arent even sent outside county lines!
nuff said
 
6:57 AM
When miao miao jumped out of QC to join this company, the physics people who heard what miao miao was doing in QC, all agreed that miao miao dodged a bullet. And then the people who touched QC before, added their own QC horror stories
 
@naturallyInconsistent god
 
this notation makes me sad ;_;
 
yeah, eyes are not happy with that
@Relativisticcucumber anyway, your latest question about BEC, im not sure what you want more than the fact that Bose-Einstein statistics favour a LOT of particles sharing the same state.
 
7:21 AM
i wish there was a mathematics for theoretical physics course
 
@SillyGoose actually, miao miao benefitted a lot from one
but even that is not at all enough. There are just way too much maths that can be applied to physics
 
true dat
i really wish i knew diffe g...but maybe i can learn enough over this summer
 
123
7:36 AM
Hello Everyone..
 
123
What is the gravitational field equation in 2D in newtonian mechanics?
 
Depends how you derive it :p
 
123
At which i can find field at point in space as vector field by magnitude and direction.
 
depends on the context
 
123
7:38 AM
like the case of earth gravitational field
 
You can just pick it to be the same as the Newtonian one, the Laplace equation : $\Delta \phi = \rho$
In which case since it is in 2D the field drops off as $1/r$ instead of $1/r^2$
 
123
$g = \frac{GM}{r^3}\vec{r}$ is one dimensional
 
The force I should say
The field drops off as $\ln r$
 
123
how to do this in two dimensional in Earth case
 
@123 what context are you working in? are you trying to describe the force of gravity on an object close to the earth's surface?
and what do you mean by 2D? are you looking at the earth as if it were a circle drawn on the page and an object drawn above it?
 
123
7:40 AM
@SillyGoose I just want to create Earth gravitational field nothing else.
 
@123 this is a vector equation. why do u say it's 1d
this eqn is for 3d
 
123
@RyderRude because it has only $\vec{r}$ not angle
what is the description of "r"
 
$r$ has 3 components, so 3d
it has the vector r, not the length r
 
123
@RyderRude Can you pls share the equation of "r" which shows 3D
 
$r = x^2 + y^2 + z^2$ as measured from the center of the earth (most likely but it is hard to tell because i am not really certain what sort of context you are working in)
 
7:43 AM
@123 can u pls first describe what this formula means to u
 
123
@SillyGoose Thanks. What do you mean by context. Can you pls tell few example of different context.
 
@123 if you are writing down this formula then it sounds like you are doing a problem in which you are considering the earth and another object in space and considering the force one mass exerts on the other
alternatively, you can be at the surface of the earth and considering the gravitational force acting on an object near the surface of the earth like you do when you consider a baseball flying through the air
 
123
@SillyGoose No , i just want to draw vector gravitational field of Earth.
 
okay so are you just considering earth in space
 
123
@SillyGoose Yes
 
7:47 AM
@123 then the formula you wrote seems appropriate
for 3D
 
123
@SillyGoose Let me animate vector field using this formula. Then i share you.
 
i think you are missing a negative sign--the gravitational force should be attractive
 
123
I do not have problem with minus sign. but $\vec{r} = <x^2 , y^2>$ doesn't work
$\vec{r} = <-x^2 , -y^2>$
It is not looking spherically inward.
 
$\vec{r} = (x, y)$
 
123
What is the problem with my gravitational field
 
7:54 AM
if you are looking at earth in 3D space, it has gravitational field $\vec{g} = -\frac{GM_e}{r^3}\vec{r}$ where $r^3 = (x^2 + y^2 + z^2)^{3/2}$ and $\vec{r} = (x, y, z)$. If you are looking at just a plane in space (so specializing to 2 of the 3 dimensions), just set $z = 0$ and let $\vec{r} = (x, y)$.
 
123
but scalar "r" doesn't help in animating field.
Pls share equation of $\vec{r}$
Is it okay $\vec{r} = <-x , -y>$
 
the force shud increase as we go toward the origin
it is undefined at the origin
 
123
@RyderRude Aaah okay.. let me again animate
 
You should be trying $$\vec F_G=\mu\begin{pmatrix}\frac{-x}{(x^2+y^2)^{\frac32}}\\\frac{-y}{(x^2+y^2)^{\frac32}}\end {pmatrix}$$
 
123
@naturallyInconsistent Aaah Thank... This is what i am searching for.
@naturallyInconsistent Perfect it worked thanks a lot
 
9:09 AM
the militaries of modern countries are something out of sci-fi when compared to the greatest militaries from hundreds of years ago
air force alone makes it a completely different thing
but in the movie Avatar, they show the old war techniques of horses and arrows winning over modern military tech in a war
but they didnt use nukes in the first movie maybe because of public backlash back on earth. in the beginning of the second Avatar, they use a nuke and win the war
 
let $M$ be a smooth manifold. let $x \in M$. i want to define the tangent space $T_xM$ as the space of all derivations over smooth functions at $x$ (with appropriate addition and scalar multiplication defined on the derivations).
I am a bit confused because wikipedia states that a derivation as $D: C^\infty(M) \to \mathbb{R}$, so it sends a smooth function $f: M \to \mathbb{R}$ to a real number.
Is that right? I would have thought that it should send the smooth function evaluated at the point $x$ to a real number, so $f(x = x') \mapsto Df(x) \lvert_{x = x'}$
 
9:33 AM
u can think of $D_{x'}$ as the map $D_{x'}(f(x))= Df(x)|_{x=x'}$. the input has to be the function $f(x)$, not the function evaluated at the point $f(x')$ becuz the latter loses the neighborhood information required to differentiate @SillyGoose
to be more precise, the input is a germ of functions
becuz the equivalence class of all functions, which hav the same neighborhood information around $x'$, get mapped to the same real number
 
i see
 
10:16 AM
Oxygen started out as poison for most of the life on earth
life persisted despite that massive catastrophe
now life has come to a point where we might start terraforming other planets after another 1000 years
 
10:44 AM
@SillyGoose it is in fact wrong. You should see it as a map $D: C^\infty_{x^0}(M) \longrightarrow \mathbb{R}$. $C^\infty_{x_0}(M)$ is usually called the germ of $C^\infty(M)$ functions at $x_0$. It is the quotient of $C^\infty(M)$ by the relation $f \sim g$ if there exists a neighbourhood $U$ of $x_0$ such that $f|_U = g|_U$
this is just telling you that the derivations at $x_0$ should only depend on the local behaviour of the functions at $x_0$
 
glS
@naturallyInconsistent sorry, stumbled about this conversation and am very confused. I know who you're talking about in MIT, but who is "miao miao" here??
 
 
2 hours later…
12:39 PM
@glS when naturallyInconsistent writes "meow" or "miao" he means "me" or "I"
 
I always thought "miao" was Italian for "meow." :P
 
It is
It's pronounced the same way
 
> “Meow” is an established spelling in English. Rossini spells it “miau” in his famous duet for two cats - but he would; he was Italian. “Miao” is the preferred French spelling.
re: Quora
cc @Slereah
Now I'm wondering about the German spelling 🤔
 
12:55 PM
@user85795 "Züfefrysvrfhrbden"
 
Thanks pal.
 
1:54 PM
What's a Cartan connection
 
2:20 PM
A connection that has a "translational" component to it
 
3:10 PM
@Slereah it seems related to solder forms somehow
 
3:38 PM
Ahh the translational part IS the solder form
Ohh it's called affine connection because of this translational bit
 
3:56 PM
8 latest consecutive questions all closed
Longest chain I've seen so far
 
hi
there was a spacecraft which recorded Saturn
these planets are hellish rubbles
i really want to be in different places of the universe, but our tech doesnt allow it
people get to be in the universe only to roam a point sized volume
 
4:23 PM
@SillyGoose all the big language of bundles and manifolds and derivations wasn't very helpful when it came to actually computing a simple derivative, there is a very deep lesson in this
 
5:13 PM
I love that bolbteppa is magically summoned as soon as there is a possibility of insulting bundles :P
Confess @bolbteppa you dream of bundles in your most gruesome nightmares
 
@Mr.Feynman oh, im sure he is admitting guilty of something
 
@user85795 miau
 
5:28 PM
anyone else get bugged out when people write the $dx$ before the integrand in integrals?
like I know what they're going for by making it look like an operator, but it just feels weird
 
I had a phase where I preferred that notation :P
nowadays I don't really care
 
real ones dont write the $dx$
 
@nickbros123 from what I recall in my shoddy calc on manifolds class, it does actually matter in some cases
 
@nickbros123 well, you kinda have to write it when you're about to do substitution or potentially work with more than one measure
 
what's really weird is this derivative notation
$d^2$ in the denominator
like again I get what it's going for, but just keep the notation conventional
 
5:33 PM
but sure, I, too, prefer the pristine efficiency of notation like $\int_V \omega$ integrating a form over a volume
 
speaking of the subject matter, @SirCumference, I recently bought the hard-cover version of spivak calc on manifolds :)
 
@nickbros123 I'm so sorry
 
@SirCumference xD
 
@nickbros123 go study biology lol
 
sad lyf
 
6:02 PM
@SirCumference Yes, it should be better written as solid angle
The power of two looks ugly
 
@SirCumference One becomes an adult when this becomes their default ;)
 
@bolbteppa I want to stay a kid forever then :P
it's just nice to have it clearly written where the integrand ends
also just looks more appealing tbh
 
only disadvantage is $\int (\dots)+(\dots)dx$ looks like it's enclosed by the integral and dx whereas $\int dx (\dots) + (\dots)$ need to use a bracket/parentheses to enclose what you're trying to integrate
 
6:17 PM
there ought to be an international mathematical committee that decides the convention for a lot of the notation
not that anyone would need to follow it, but we could deem any deviation from it as nonstandard
well then again, I doubt physicists would follow the conventions mathematicians establish in any case
 
Don't you think standards detract from the beauty of the art that is mathematics? :P
 
I do not, in fact :P
 
If one is to believe beauty is immutable and the way one perceives it doesn't matter 🤔
when one "understands" something in mathematics is like any other evocation :O
I was kinda tongue in cheek about it because I don't really think of math as "art" like most mathematicians do but now Idk
like how a piece of art can evoke deep feelings in some regions of the brain, math can do the same just in a different way..but one is emotional and the other is logical. I'm not convinced you can call them the same thing (calling math art)
 
$\int dx f(x)$ has always looked more badass, when you've suffered through qft you've earned the right to write this notation
 
I agree. But when everyone starts doing it, it becomes less cool like most things..
 
6:39 PM
@bolbteppa I realized I am a man by now when yesterday I wrote an integral with the measure after the function and I barred it to have the measure first :P
It's also much easier to read, so you know the variable first. It's similar to the Spanish ¿ (although we don't put the measure at the end too :P)
I can't let myself do that now though, not until I take measure theory. My prof would definitely fail me if I wrote $\int d\mu f$ :P
 
I remember purposely not doing it until I got to qft
 
What is astonishing to me is that somebody once mentioned it in the MSE chat and they were like "no one uses that notation"
I got a similar feedback when I used $\frac{1}{A}$ for the inverse operator and there was no way to have them accept it :P
 
@bolbteppa I don't trust the quantum physicists with notation after they started putting arrows on top of norms
or in some case, boldface
like wtf sakurai
well actually that's a bad example
but i swear it's something he does and it drives me nuts
 
I don't understand the problem :P
 
what's wrong with that notation
 
6:45 PM
Physics notation is just efficiency, when they write $|x|$ you know they mean a 4-vector (or an $n$-vector depending on the context)
 
@ACuriousMind I chose a terrible example in the heat of the moment
this one's better
 
I love the way you panicked right after sending that pic
 
like why is $J^2$ bold and yet $J^2_z$ isn't
$J^2$ isn't a vector
 
@SirCumference oh you mean $\vec{v}^2$ instead of $|\vec{v}|^2$?
 
6:47 PM
again, nothing wrong with that
since $\vec v\cdot \vec v = \lvert \vec v\rvert^2$
 
Because $\vec{v}^2$ is shorthand for $\vec{v}\cdot\vec{v}$
 
it's just that it's bizarre to write norms (scalars) in the same fashion as vectors
 
and of course $\vec J = (J_x,J_y,J_z)$ is a vector (operator)
 
Like this isn't a thing that most people do outside of QM anyways
Usually in classical mechanics people would just write $J^2$ without the boldface
 
So, if you want to nitpick: $|\vec{v}|^2$ is by definition $\vec{v}\cdot\vec{v}$, while $\vec{v}^2$ is just a shorthand for the latter
 
6:49 PM
and more generally mathematicians don't even use arrows or boldface when writing vectors in linear algebra
@Mr.Feynman Most people would just write $v^2 = \vec{v} \cdot \vec{v}$
I mean I guess it doesn't really matter, but it just feels weird to see arrows on top of scalars
 
@SirCumference in that case they mean $|\vec{J}|^2$ of course but it's potentially confusing to use square without either $|\cdot|$ or $^2$
 
@Mr.Feynman well that is true
 
@SirCumference that's because in classical mechanics there is always a number $J$ - the QM intro text is emphasizing that while the operator $\vec J^2$ is a scalar operator, there is no scalar operator $J$ that you could square
 
@ACuriousMind I mean I feel like that really only needs a paragraph of explanation
not modifying the whole notation in the book
 
sure there's other ways to do that, but it's just a choice of convention
 
6:52 PM
I guess so
 
I love also that the discussions about notations are the most heated up :P
 
there's nothing wrong about doing this, and it is internally consistent
 
like most things, how we feel about convention is arbitrary
but some things just stick out like a sore thumb to me
@Mr.Feynman that is true lol
 
All physicists know that conventions are conventions and yet they'd be ready to kill to defend their favorite one
4
 
quite so
 
6:54 PM
@Mr.Feynman for me it's more that I rag on physicists for not doing math properly all the time, so I feel some need to defend them when they're just being a bit silly about notation :P
 
I wonder what physics would look like if they followed all the advice of mathematicians
well I guess just mathematical physics
but I'm curious if it'd really make the major topics that much harder to learn
 
@ACuriousMind we kind of are on the same boat, yet
it is time to torture ACM now. Let's switch to passive vs active stuff
 
Active vs passive can get you into real trouble unfortunately
 
@SirCumference still studying simple pendula
@bolbteppa Most definitely, I hate discussing that stuff
Every time I manage to get myself confused
 
7:52 PM
@bolbteppa true it’s not so useful for computing things…but one doesn’t really need to know much at all to compute things. The “abstract nonsense” is my preferred way of organizing information in my head :P
It is hard to visualize the abstract holonomy of a connection…and i prefer to think in words anyways. “Abstract nonsense” also lets you recognize things in the future with little additional effort
it also just answers questions that people maybe are okay with glossing over but which seem pretty fundamental. like why should the connection 1-form be lie algebra valued? and the lie algebra of the structure group no less. well the answer is answered immediately if one just constructs it, and the construction is quite natural
@SirCumference i like it :D it makes me not forget to add the measure in the first place lol
a lot of stuff even in the abstract nonsense just seems to be linear algebra anyways...
 
8:16 PM
How does an LLM like chatgpt or any of the other ones correctly analyze problems?
I know at the end of the day it's just a brute forced non-intelligent piece of code but it's weird how it can do moderate computations and problems
 
 
2 hours later…
10:36 PM
@SirCumference Have you tried reading a quantum mechanics book written for mathematicians?
It's impossible to decipher what is actually going on lol
 
11:03 PM
what do you mean ;)
 
11:17 PM
lol precisely
 
11:47 PM
let $P$ be a principal bundle. define a connection on $P$. i am trying to show the equivalence of the horizontal subspaces defined by the corresponding connection one-form and the horizontal subspaces defined by the definition of a connection.
I should be able to ignore definition 10.4 and from the listed properties in the definitions show that $X \in H_pP$ iff $X \in \ker\omega_p$
Nakahara proves one direction in the text, so I am attempting the other direction and got to this point.
However, I am not sure how to show that $(10)$ vanishes
 

« first day (4914 days earlier)      last day (30 days later) »