
Two things  no three.
ballyhoo, the diagrams you copy/pasted from pmb's website are themselves copy/pasted from a book by Bernard Schutz which is almost certainly copyrighted. Probably not wise to post them here.
I am ashamed to admit that, in talking about set functions, I forgot to mention something we shall need.....
Suppose be sets and that and that . Then the image point . Then by applying the function to I may have that . But this notation is rather cumbersome, so one writes .
This is called "function composition" which we can think of as the rule "do f first and then do g", (i.e. read righttoleft) but it is better to regard this as the single function .
So to return.....
We have that the homeomorphism , so that, for some that .
We also have that such that . So, again assuming the "reality" of our point, we need to find a relation between and .
We are talking manifolds, right? And these are not in general Euclid "flat", so we may not assume a linear relation between these two ntuples. That is, one may say that.....
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.
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But notice we have n independent functions, each of which uniquely determines each . In this circumstance it is customary to "economize" on symbols and write, say,
and so on, as above. (I don't like this, but it is standard notation).
I now need to introduce another standard bit of notation which I DO like, but which may need some explaining.
Suppose for te above I write where , I intend that the in this equality be taken one at a time, in succession, if you like. Whereas I intend the argument to treated as an ensemble. In fact we can take this as a rough ruleofthumb.......
Indexed elements are to be taken one at a time, unless they are enclosed in some sort of bracket, where they are to be treated as an ensemble. Moreover we may have occasion to select a single element from an ensemble  which may be a tuple, or a set or whatever.
So suppose I write the set as ( an ensemble), I may want to select the ith element in the set and say "for some (or even each) soandso is the case"
I am not sure this is at all clear, but it is Very Important you understand index notation.
Somebody help me out!!!
Last edited by Guitarist; August 30th, 2012 at 02:53 PM.
Well you guys are no fun!
Don't you want want to hear about arbitrary coordinate transformations, tangent spaces, cotangent spaces, vector and covector bundles, vector and covector fields.......
Unless I was preaching to the already converted, it seems not. Ah well
Well I'm still here, but I'm still digesting the material from a little ways back.
I've been playing catch up also, i'm still a little puzzled by the notion of a topological space being compact. (but that's for another thread and another day) As for everything else i think i'm still hanging in.
The index notation you outlined seems simple enough, although i do share your dislike of the customary way in which symbols are economized.
Hi Wallaby
If you want to start a new thread I can help you with any issues you may have with compactness
So, disregarding the possibility that this thread has come to his natural end, let me start up again.
First this: Let's not get too hung up on compactness. We can think of a compact space as one that is as "large" as it needs to be but no "larger". Of course, for a general top.space we don't quite know what "large"  or "small" for that matter  means. But it will not be important in what follows.
Let me try to give another piece of intuition: Recall we agreed that a manifold "looks" locally like some , and I wrote this as, for the open set , that the invertible mapping defines the image of the point to be where the RHS is some point in .
Now any comes equipped with a set of coordinates, so very roughly speaking we can think of this as "placing a transparent coordinate grid" on such that is "transferred" to a point in with "coordinates" .
This is fine as far as it goes, but sad pedants like me will tell you that the ntuple is not strictly speaking a set of coordinates  but we can make it so as follows.......
Notice that, for any that n times. This is called the Cartesian product of sets. For simplicity consider the SET whose elements are, say, .
Now the Cartesian product comes equipped, in this case, with a pair of maps called "projections" such that , and the are called respectively the first and second coordinates. So, if we gather these into a SET we have as a set of coordinates for our point in the Cartesian product. It is important to observe the different parentheses here.
So this is nice  for the image point we have a set of coordinates . BUT it is not nice enough,,,,,,
Wanna know more? Stay tuned or, for preference ask questions. I warn you  if this was tough going, what follows is not for the fainthearted
i'm still on board, for now.
Guitarist, is your notation standard here?
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