Got the theorem, having trouble with the proof... [SOLVED]
Hi all. OK, so I am trying to prove a theorem that I have for some time been just using as-is. Long story short, it occurred to me that I needed to prove it. So, I have almost done it, but am stuck near the end. The theorem is...
Having been put through the workout that was Spivak, as well as Bishop and Goldberg, Warner, de Rham (actually quite good, albeit difficult), and more, I can honestly say that all of these make a poor choice for a first course on differential geometry. While Spivac keeps things... simple(?), I...
I would probably advise people here to quit wasting time on this, as it is not for any sort of class per se, but rather a (weekly?) competition put out by some sort of RPG-ish gaming website, http://www.neopets.com/games/conundrum.phtml Not that there is anything wrong with doing math...
Correct, depending on how you define your charts. However 'almost' enough is not enough when in comes to math. Engineering on the other hand... :smile:
OK, so I got the picture correct, so I may as well get to work. Wish me luck!
As for the question of two charts, consider that this dumbell is topologically equivalent to S^2 , so we need only consider that shape. On an intuitve level, to say that a manifold can be covered with one chart...
OK, so to make sure I have this picture correct, we can make this construct by drawing a circle of radius R, and then drawing another copy of the same circle, centered at a point at a distance of |2R| from the center of the the first. Then, at a point 'above' the cusp, we draw a smaller circle...
Well, I would be willing to take stab at this, except that I cannot figure out what your construct is supposed to look like--I don't understand what you mean by "the circles osculate". It's simply a matter that I have not come across this term before, and thus need a little elaboration... If...
Personally, I found Churchill to be a bit tedious (sorry, just my opinion). I found "Complex Analysis" by Lang to be very good--especially if you take the time to do the excercises. It get's to the point of WAY beyond high school, but it builds slowly, and you can always stop once you hit the...
Maybe. I was looking at your problem last night, and I think I should be able to have something for you by this evening or tomorrow morning (party tonight, so no guarantees for today...)
However, while we're on the topic of soliciting responses, do you have any ideas about my question of a...
Oh, I don't know. If I can figure this stuff out I'm pretty sure that it cannot be that difficult! Thank you, though, for the interest--it's a good opportunity for me to practice.
That said, let us begin.
Let \mathcal{M} be a manifold, and let \mathcal{U}\subset\mathcal{M} . Define a...
An update:
Well, the first problem has been taken care of. The key was to find a text that treats Riemannian geometry from a somewhat elementary standpoint--numerous examples were found in various continuum mechanics texts. If anybody is interested, I can post the result here.
Now, the...
In your example of the torus above, you have stated that "the same shape can have many inequivalent metrics", but I don't think this is quite correct. For example, in the above, we could denote the metric that is inherited from Euclidean 3-space as g_{ij} , and the one corresponding to the...
What you are calling a surface patch is (I think) more commonly referred to as a chart, so with that in mind you should be able to be able to google any number of tutorial style papers on the subject. As for a text, I personally have found the book "Lecture Notes Differential Geometry" by S. S...