Recent content by majutsu

According to my text, a manifold should be 1) Hausdorff (that is t-2 separable, so there are disjoint open sets which are neighborhoods for any two points x and y), 2) locally euclidian (that there is a neighborhood U of a point x that is homeomorphic to an open subset U' of Rn (the RxR...xR...

Thanks everyone for great suggestions and ideas. I am still reading . . . :) I actually got detoured, because I wanted to brush up PDE and the fine points of advanced calculus before DG, so that's where I'm at the last 2 months or so.

This is a really awesome thread. Thanks for all who contributed. I learned a lot. Mathwonk, you especially deserve kudos for the post discussing calculus and the PhD syllabus, as that personal tidbit really brought the concept of "deepened understanding" to life. Thanks a lot.

I see the problem. It is not a misprint, just poor language. The original text says (paraphrased) a homomorphism of G into H as F(G,*g) ---> (H,*h) and F(g1 *g g2)=F(g1) *h F(g2) Then the book reads, "if (iii) every h in H is an image we have a homomorphism of G onto H." This is...

Consider the cyclic group G={a,a^2,a^3,...a^12=u} and its subgroup G`={a^2,a^4,...,a^12}. My book says that the mapping a^n ---> a^2n is an homomorphism of G onto G` (this seems true) and that X: a^n ---> a^n is homomorphism of G` onto G (this seems to be false to me, a misprint) A...

Actually, I get it even more . . . You can define a cut C as {x in Q:f(x)<a} where x is some function of rational numbers (x,+,/,-,^). C' is {x in Q:f(x)>=a}. In rational cuts, like {x:x<3} the cut point can be set either way, here on the right half, and either half can define the cut. In...

Hurkyl, are you saying that we need only define one solution to s^2=2? In that all cuts of s and say s' would be in the same equivalence class anyway? I guess that makes sense, finally. I think I might have this now . . .

Both of these posts help. I think what confused me is that my book started with rational Dedekind cuts, and sketched out the rest leaving it (infamously) "up to the reader". One question remains though, if the cut is rational (in other words if we start with Q and try to derive R), my book...

Can someone explain to me how dedikind cuts derive the real system from the rational number system? I understand how the cuts derive an algebra with a defined addition and multiplication, etc. How do we jump from this algebra on these cuts to assuming the existence of the irrationals? I guess...

thanks, all. Good stuff. RDT2 thanks for the links.

Thanks for all the responses. Hi Samoth! I love the Schaum's especially for Linear Algebra, and will probably get the differential geometry book, although I hear it's only classical differential geometry. Similarly, they say Kreyszig's book, with the coordinate p.o.v. is limiting in the...

What are the best books for learning differential geometry well? Any recommendations appreciated.