If I ever gain influence on the educational system of my country (which will never happen), the first thing I'd do is make sure degrees aren't taught in elementary school
Look back at what you had before. It was of the form $a Y_1+b Y_2 = c Y_1 +d Y_2=1$ where $a,b,c,d$ were functions of $x$ and $Y_1,Y_2$ are $f(x/2)$ and $f(2/x)$ respectively.
anyway. what would you classify a horizontal line of fixed points as (ie stable, unstable, saddle etc) when there is no motion along that line, but above the line trajectories move right and below it trajectories move left
what would you classify a horizontal line of fixed points as (ie stable, unstable, saddle etc) when there is no motion along that line, but above the line trajectories move right and below it trajectories move left
@SoumyoB you say assignments, so presumably you're taking a class, and presumably you're learning from source material, so you have personal notes, lecture notes, or a textbook to look at, no?
fine. but that's still equally useless. following arctic's suggestion, you've taken $(4/x)A-B=1$, written $B=(4/x)A-1$, and then substituted to get $(4/x)A-((4/x)A-1)=1$.
which is true---but again, useless. an equation of course satisfies itself.
@Balarka The main problem with your new avatar is it's not immediately recognizable like your green one was. I think I have a similar problem, but I guess it's easier to say "oh, Mike's speech bubble is here"
Wells' book is the best source. Warner is ok, but I think his proof sort of half-asses it. I learned a lot of the standard elliptic regularity theory from a PDE class.
Like, his proof is correct, but if you're going to do the hard analysis you may as well do it on Euclidean space and not do the tricks he uses to work things out for compact manifolds only.
@Ted I still think that if you're going to go through the trouble to prove the various estimates, you should just do it in Euclidean space. Sure, now you need to use the word "coercive form". Whoop-Dee-Doo.
@Ted I still think that if you're going to go through the trouble to prove the various estimates, you should just do it in Euclidean space. Sure, now you need to use the word "coercive form". Whoop-Dee-Doo.
@Danu: We had talked plenty and he knew what I did. But in the case of my one doctoral student and plenty of MA students I've advised, I spent literally days reading, criticizing, rewriting things, complaining about errors, complaining about style, etc.
@Ted Yup. I'm not really debating, this is just opinion. I think I learned the "idea" a lot better when I did it on $\Bbb R^n$ and when I did it via symbols. I think Warner's proof is unenlightening.
Karim: A lot of people don't agree with me, but if we're going to call it differential geometry, I want to see some sort of metric or some sort of connection.
@Ted I think there are other kinds of geometry, but of course I agree that manifolds by themselves are not geometry. Or else you'd have to call me a geometer.
If all you do is the structure of differentiable manifolds, vector fields, flows, tensors, differential forms, Stokes's Theorem, this isn't what I consider geometry. It's foundational stuff on manifolds.
I understand, @MikeM, but symplectic geometry and projective geometry have their own structures that make it more than a differentiable manifold. And a lot of more modern projective geometry is about projective connections :P
Well, the parametrization is awfully yucky without the double-angle observation. I didn't try differentiating.
In my undergrad diff geo class when I did the hyperbolic plane, I computed the geodesic curvature of such circles and gave an exercise (that only a few did) to show that hyperbolic circles are in fact Euclidean circles with different centers. That's not at all obvious.
