Are you kidding? You're prime Grand Vizier material. You'll be a natural at cackling, having people thrown in the scorpion pit and wearing a turban with a point in the middle.
a friend of mine did like a lot of self-studying in high school and then took diffgeo for graduate students in the first semester and got the highest possible grade
well if I were you I'd just take the ones you already know anyways and get them for free and in the meanwhile attend courses that you don't know everything about yet
The one time my father called me for help with his homework, he was stuck because he hadn't had an equivalent course. After one change of variable we were able to establish that he had the complementary error function and then he could use a table ...
@Danu Uh...not sure if I'm looking at this wrong, but the red lines in the spiral do not seem to me to project down onto the red line in the circle, they would be more like the "left" half-circle than what's drawn there
hi guys! is there a difference between the "non-coordinate basis" of a Riemannian/Lorentzian manifold and a "locally flat" basis? Nakahara defines the "non-coordinate basis" like this, which sounds to me exactly like locally flat coordinates..
@Bass The locally flat basis is a non-coordinate basis, but not every non-coordinate basis is locally flat (non-coordinate just means it doesn't arise from a choice of coordinates, nothing more)
@Bass Yes, what Nakahara writes there is the locally flat basis, and really I'm not sure if there is use for non-coordinate basis which aren't locally flat
@ACuriousMind ok cool. Have you read (parts of) Nakahara's book? just want to know how mathematically rigorous it is.. to me it seems pretty rigorous, but still, it's aimed at physicists and covers pretty much in 500 pages, so I'm not sure
@Bass I learned most of it from lectures (and by osmosis, perhaps, as I can't recall where I learned some things), I can count the number of textbooks I've actually read on one hand
Would anyone here be up for attempting to answer this question on WB? I don't mean to plug the site or the question, but I'm betting that it won't get answered anytime soon.
Assume that 437 days is a reasonable limit for how long a human can endure constant-velocity space travel. Proxima Centauri, the star closest to our Sun, is 4.24 light years away from Earth. If you wanted to fly to Proxima Centauri within the 437-day limit in a rocket of mass 2.00×10^6 kg , how m...
@Danu The motivation for the germ and equivalence class definition that physicist's tangent vector definition only helps for manifolds embedded in $\Bbb R^n$.
What the hell is your tangent vector definition. I thought you were talking about the perp of the gradient vector pointing away from the manifold at that point.
I will read the relevant section you have mentioned tomorrow, but I am trying to expressing my opinion that a question should be reasonably self-contained.
@BalarkaSen I sympathize, but I do think the context has been elaborated on sufficiently---and I think most people would agree with me. The point of the question is not the precise technicalities behind everything, but rather to get a historically oriented POV on the developments that led to these definitions. Anyone able to answer the question will know the definitions already.
Usually, by tangent vectors to charts, I thought you meant the plain old tangent vectors in R^n because Guillemin-Pollack's definition of a manifold is a manifold embedded in R^n.
I can tell you why the derivation interpretation is useful, but I don't think that'd be the original motivation behind it.
You can generalize derivations to produce tangent spaces of affine algebraic varieties and affine schemes. It's a very sheaf-theoretic thing - a linear map with leibniz from stalk to $k$.
My picture is just a sketch of the universal cover. The real thing looks much much messier.
It's $\Bbb R^2$ with a sphere attached at each integer lattice, and with each sphere attached an $\Bbb R^2$ and for each of those $\Bbb R^2$ again a sphere attached at each integer lattice point and so on.
QUOTE: A grope is an infinite construction using disks. If you have a map from a 2-disk (circle with interior filled up, yes) to a 4-manifold, it can have a lot of singularities. Now around each singularity, pick loops based at the singular points and similarly attached 2-disks. Those in turn have singularities. Keep attaching them. You'll end up with an infinite tower of singular disks. ENDQUOTE
i.stack.imgur.com/x2qJM.png
I don't understand: We have checked that my understanding of continuous map between topological space is correct
And we know there is no way to perturb to remove self-intersection for maps $D^2 \to M$ where $M$ is a 4-manifold, although it is possible for 5 and higher dimensions.
So this is what you do - keep pushing the singularities off the frame. To infinity.
Well, gropes are some kind of generalizations of Casson handles. I don't know much about this.
I can only tell you that a Casson handle, when thickened, is homeomorphic to $D^2 \times \Bbb R^2$. So it effectively takes a singular disk and results in an embedded disk (no self intersection) with the same boundary.
