I think that Didier's objections, while they do point out details that were left out, were mostly misinterpreting my argument. I wasn't making the errors that he was indicating; I was leaving out details.
From the history of your answer... somehow I think there's a point where you really cannot and should not pack everything into an answer. It should probably already be a note/paper/something...
I have to go out tonight, and then do a lot of backed up work. I may be inactive for a while. Not that I have been very active recently. The end of the holidays is depressing.
@JM Yes, I agree. I was trying to point out the method of derivation of the formula, but as Didier points out, a proof was requested.
The method, which is an outline of a proof, is pretty easy to follow. When all the details are added to make a proof, it is a bit unwieldy and hard to follow.
FWIW: I can still read the current version of your answer, but it now takes some stamina on my part... and then the reason it doesn't take much stamina is that I've seen the first version.
Byron's answer is slick and answers the question asked. However, I don't think that it offers much motivation. Before my mods, I think my answer showed how one uses the ideas of stationary phase (when brought to the real line).
@Srivatsan I hope that Didier does not turn out to be like Bill Dubuque was to me on sci.math. I got no breaks and he seemed to push hard on my answers.
I also don't want to start up bad relations here. Bill seems to be accepting my methods better here. He even used a proof similar to one that I had used on sci.math and he had ragged on me about.
I was amused.
Combinatorics and number theory were the places we clashed mostly.
I liked to prove things using Bezout, because it ruled out the circularity that sometimes creeps in when using unique factorization (Fundamental Theorem of Arithmentic)
I mean, when I first became interested in math I looked around for relevant communities and hence lurked on sci.math for a while. And I became convinced that it was a mess.
@Srivatsan Some of the titles are related. The last user says in his one question: "As I have in my previous questions (sum of square numbers (difference way) - algebra precalculus) "
Imagine that I am at a fixed altitude z and I am staring at a mountain (the function) and I swing a huge straight bar from where I am until it crashes against the mountain
You're saying pseudo-tangent and worrying that you can't calculate a derivative of this thing (or that it might not even be differentiable?). I'm not sure that I, personally, can say anything interesting in that case.
I want to find the impact point against the mountain
I just discovered that I can find the derivative, btw
I say pseudotangent if only because I don't know the proper jargon for such an intersection, as it may not be a true tangent but rather a point of first intersection or something
This would be a little messy, but maybe you could still use what JM does in his answer. You look at the various planes passing through your point P (containing the vector that goes straight up from P?) and do the 2D analysis there.
I'm obviously no help with this. The main site might be, although you would want to be able to articulate this well.
Don't mean to create a new thread, but how do you study if there are no answers at the back and no solution manual. How do you know if you are on the right track? :)
Well, I think there's an important distinction between "I don't know how to do this" and "I thought I solved this, but there's a gap or missing justification that I don't see."
I also think that if you're doing exercises out of a book and you can't do the majority of them, then probably something has gone wrong. If there are just a few that seem very hard, then you think about it off and on and then maybe you ask someone. This site is good for that.
But you do have to get used to it. Most books don't have solutions and it seems like a minor miracle to find a book that has a solution manual and is any good.
There's Spivak, for example. I can't think of many others.
There's a whole family of tangent lines, and they comprise a plane. Unless you put in further constraints, we can't have a unique line.
@user1123950 Right, the man can be "anywhere" at the foot of the mountain, yes? That's exactly why your problem is underdetermined; you haven't said where the guy's swinging his bar from.
The intersection of this with your surface is some curve described by z_0 = f(x, y).
Say (x, y, z_0) is on that curve. You can write down the line between that point and (x_0, y_0, z_0). You can also write down the equation for the tangent line to the curve at the point (x, y, z_0).
You more or less want the slopes of those two lines to match up, right?
(1) I can draw two lines at (x, y, z_0). One is the tangent line. The other is the line going through (x, y, z_0) and (x_0, y_0, z_0). I want these to coincide.
So I mean we've got a "true" tangent line on the mountain that may or may not intersect me, and a direct-connect line from me to that point on the mountain, yes?
That's what I mean. So there's more work involved there.
You could probably compute the angles and take the "leftmost" and "rightmost" ones.
At any rate, it would be good to work this naive approach out. I might be able to talk more later, but chat is so lively that maybe you'll have hammered this out by the time I get back.
Dogs are great but the upkeep is a little nuts sometimes.
@BenjaminLim No I don't. I guess it wasn't really a question and it was attracting too much attention while not being constructive at all. You don't have to give a reason for voting to delete a question.
Have fun. I always forget that it's summer over there (I wish it were summer here, too :))
One way that might be a worthwhile answer would be to look at harmonic forms. Unfortunately I forgot about the details and I currently don't remember where I saw them (it might have been in a Séminaire Cartan). Basically you can compute the cohomology of any symmetric space using $G$-invariant forms without taking cohomology at all.
The point is that harmonic forms are automatically closed.
The best way I can describe it: A man sits at arbitrary point (x,y,z) and there is a mountain nearby, describable with z = f(x,y). The man takes out a bar, holds it parallel, and swings it counterclockwise until it connects with the mountain at the same level z. I'm trying to find the point of connection.
@user1123950 Let's call the point $(x_0, y_0, z_0)$. Now you need to determine the curve described by $f(x,y) = z_0$. If you choose a coordinate system in the horizontal plane your problem reduces to finding the tangent through the point to a function, which is pretty easy. The difficult thing is finding a good description of the curve implicitly described by $f(x,y) = z_0$.
@DylanMoreland Well, yes, but the difficulty is to find the point on the curve through which the tangent line is supposed to pass. I don't know how to do that without actually solving the implicit equation.
Aha. I was responding to something he said about setting up the tangent line.
If you have a point in hand, I think what I said above can tell you whether it's good or not. I just saw Steven's answer and that seems to be a nicer way of doing it, especially if one wants to shove this into a computer in the end.
He addresses the implicit stuff in passing at the start. I wonder what he would have to say about it.
There are general results like the implicit function theorem. I use that from time to time. I don't really know how to turn it into a calculation, though.
Actually, now I don't know what to make of his comment at all. I thought he was discarding the case of a curve defined by f(x, y) = 0, but he instead talks about y = f(x). Which is pretty simple.
So I don't know if he has any ideas on that at all.
Ah well. Maybe you'll find something. The real difficulty here seems to be: if I have f(x, y) = z_0 (which may as well be f(x, y) = 0) then how can I solve for y or x or just get some way of writing (x, y) as a function of some single variable?
The implicit function theorem can guarantee you existence in many cases, but you want something effective.
<rant> These constant minimalistic edit suggestions really go on my nerves. For heaven's sake: if there are three lines to edit it isn't to hard to do it in one go. If you can't do it, just leave it be </rant>
yes, but in the end you're taking a countable intersection. If you look at $G_n = (0,1/n)$ for example: any finite intersection is empty while the countable intersection is empty.
@AsafKaragila Rule 1: take at least one minute before jumping in and asking random nonsense. Rule 2: you're only allowed to ping two users within a span of 10 minutes, unless there's good reason to ping more. Rule 3: drawing attention to a question on the main site doesn't count as good reason.
@BenjaminLim No, it's not that (that's a remark one should be aware of, though). If $G_1 = (-\infty, 1)$ and $G_2 = (0,\infty)$ and $E_1 = (0,1)$ then...
