Topic Choices (Real Analysis)
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James Crook
Jun 26, 2011, 8:04:01 AM6/26/11
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Kelsey,
I see the work you've been doing on the real analysis page:
The book you've referenced has these topics:
The Real Numbers
Sequences and Series
Basic Topology of R
Functional Limits and Continuity
The Derivative
Sequences and Series of Functions
The Riemann Integral
Additional Topics
Planning of time: We should be aiming to have them all covered by mid August even though the intern program runs until end of August. That would mean working through exercises on these topics to an agreed schedule. It will be good though to talk about the ideas and where they come from ahead of reaching them in the course itself. For example, it's good to have thought about the behavior of these functions
sin(1/x)
and
x sin(1/x)
near to x=zero well before reaching chapter 5. And actually, section 8.3 is the reason 'why' we do real analysis, so although it comes right in the last chapter knowing about it will help.
I felt you were a bit hesitant about taking on this challenge - so I want to check you still want to do it?
--James.
Kelsey
Jun 27, 2011, 2:35:26 AM6/27/11
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Yeah, the past week's been busier than I anticipated. I'm doing some
reading at the moment. Regarding the exercises, should I pick out
specific ones to do beforehand or just go in and see what I can do?
reading at the moment. Regarding the exercises, should I pick out
specific ones to do beforehand or just go in and see what I can do?
James Crook
Jun 27, 2011, 9:51:45 AM6/27/11
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Where are you getting exercises from?
You should be using exercises to check/confirm your understanding, so after seeing/reading a definition of continuity, do related exercises. It is not necessary to decide in advance which exercises to do - because by the end of the course you should be able to do any of the exercises, if challenged. So if you have read up to chapter 3 and don't know how to start a particular exercise in chapter 3, ask me. Don't worry if a question you have for me 'seems stupid'. I want you to learn all the material as easily and efficiently as possible - and that can mean 'asking stupid questions'.
We'll track not just that you have read/watched a lecture, but that you've checked your understanding through these exercises. It's fine for you to pick the exercises that most efficiently help your progress. That means you also need to be aware if there are certain exercises you can't do - and ask. I may also pick an exercise for you at the time, when I want to check that you know something or diagnose what it is you don't know if you are getting stuck.
-----
I would like you to fill in the definition of continuity and of uniform continuity (in a for all epsilon there exists delta such that.. manner) on page http://onlinemathcircle.com/wiki/index.php?title=Quantifiers_don%27t_commute
Bonus: If you can write a few words that convince me that you understand why those two definitions of continuity are different that would be very good indeed, because it means you are already way ahead of where most people would be understanding the details, when starting out on an analysis course.
-----
We need to have some way to track progress.
I've created a page
http://onlinemathcircle.com/wiki/index.php?title=Real_Analysis/Progress
with some of the topics on it. As you make progress add subtopics and make very brief summaries for each subtopic. Up to 3 sentences for each. That way I can see that you are making progress.
--James.
You should be using exercises to check/confirm your understanding, so after seeing/reading a definition of continuity, do related exercises. It is not necessary to decide in advance which exercises to do - because by the end of the course you should be able to do any of the exercises, if challenged. So if you have read up to chapter 3 and don't know how to start a particular exercise in chapter 3, ask me. Don't worry if a question you have for me 'seems stupid'. I want you to learn all the material as easily and efficiently as possible - and that can mean 'asking stupid questions'.
We'll track not just that you have read/watched a lecture, but that you've checked your understanding through these exercises. It's fine for you to pick the exercises that most efficiently help your progress. That means you also need to be aware if there are certain exercises you can't do - and ask. I may also pick an exercise for you at the time, when I want to check that you know something or diagnose what it is you don't know if you are getting stuck.
-----
I would like you to fill in the definition of continuity and of uniform continuity (in a for all epsilon there exists delta such that.. manner) on page http://onlinemathcircle.com/wiki/index.php?title=Quantifiers_don%27t_commute
Bonus: If you can write a few words that convince me that you understand why those two definitions of continuity are different that would be very good indeed, because it means you are already way ahead of where most people would be understanding the details, when starting out on an analysis course.
-----
We need to have some way to track progress.
I've created a page
http://onlinemathcircle.com/wiki/index.php?title=Real_Analysis/Progress
with some of the topics on it. As you make progress add subtopics and make very brief summaries for each subtopic. Up to 3 sentences for each. That way I can see that you are making progress.
--James.
James Crook
Jun 27, 2011, 4:56:41 PM6/27/11
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Kelsey,
Notice that epsilon plays exactly the same role in each, it is only delta that behaves differently. In one it is uniform for all x. It depends only on epsilon. In the other it is a function of x as well as epsilon.
It may seem like the different kinds of continuity are a pernickety detail, but it's incredibly important.
I've also added a formulation on this page:
That makes it a little clearer that this is all about the order of the quantifiers. The two statements are identical except for the order.
When you have a chance, let me know what you are using as your source of real-analysis exercises. I'd prefer online example sheets if possible.
--James.
James Crook
Jul 1, 2011, 5:42:23 PM7/1/11
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Kelsey,
I've added a tracking page to track progress on this course on the wiki:
As you make progress I would like to see you add to the wiki, to make it clear which of these topics you've understood.
Do you know the proof that square root of 2 is irrational? This will help us get through one of those topics.
Also, do you have a copy of a book on real analysis such as Abbott to work from?
This IS a first year university level course, and I am assuming you'd like to complete it in the study group time (i.e .by end August). If that is not important to you then we can go at whatever speed you like. If we do that you will get a flavor of what real analysis is about - though there's detail that you won't have 'got' by the end of August.
--James.
James Crook
Jul 6, 2011, 5:01:09 AM7/6/11
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Kelsey,
Thanks for the new page
http://onlinemathcircle.com/wiki/index.php?title=Irrational_numbers
I'd like you to add to it a proof of \sqrt(6) is irrational, and an explanation of where the proof beaks down when you try to use the same method to prove \sqrt(9) is irrational. This way you'll have looked at the proof from more perspectives - which is good.
--James.
Thanks for the new page
http://onlinemathcircle.com/wiki/index.php?title=Irrational_numbers
I'd like you to add to it a proof of \sqrt(6) is irrational, and an explanation of where the proof beaks down when you try to use the same method to prove \sqrt(9) is irrational. This way you'll have looked at the proof from more perspectives - which is good.
--James.
James Crook
Jul 8, 2011, 2:28:51 PM7/8/11
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Kelsey,
Thanks for the update. Those proofs/explanations look fine.
What topic from:
do you want to tackle next and how can I help, if at all?
Also could you say in your own words what you think Real Analysis is 'about'? For example if I'd instead asked about trigonometry, "trigonometry is about triangles, the relations of angles and lengths of sides, and particularly building up results about arbitrary triangles from right-angled triangles. It also studies the properties of the cos and sine functions that arise in relating sides and angles."
--James
Kelsey
Jul 8, 2011, 4:24:30 PM7/8/11
to L2learning
I guess I'd say that real analysis is about functions and sets of real
numbers and the properties they have, like continuity and
connectedness and all that.
On Jul 8, 11:28 am, James Crook <james.k.cr...@gmail.com> wrote:
> Kelsey,
>
> Thanks for the update. Those proofs/explanations look fine.
>
> What topic from:http://onlinemathcircle.com/wiki/index.php?title=Real_Analysis/Progress
> do you want to tackle next and how can I help, if at all?
>
> Also could you say in your own words what you think Real Analysis is
> 'about'? For example if I'd instead asked about trigonometry,
> "trigonometry is about triangles, the relations of angles and lengths of
> sides, and particularly building up results about arbitrary triangles from
> right-angled triangles. It also studies the properties of the cos and sine
> functions that arise in relating sides and angles."
>
> --James
>
numbers and the properties they have, like continuity and
connectedness and all that.
On Jul 8, 11:28 am, James Crook <james.k.cr...@gmail.com> wrote:
> Kelsey,
>
> Thanks for the update. Those proofs/explanations look fine.
>
> What topic from:http://onlinemathcircle.com/wiki/index.php?title=Real_Analysis/Progress
> do you want to tackle next and how can I help, if at all?
>
> Also could you say in your own words what you think Real Analysis is
> 'about'? For example if I'd instead asked about trigonometry,
> "trigonometry is about triangles, the relations of angles and lengths of
> sides, and particularly building up results about arbitrary triangles from
> right-angled triangles. It also studies the properties of the cos and sine
> functions that arise in relating sides and angles."
>
> --James
>
> On 6 July 2011 10:01, James Crook <james.k.cr...@gmail.com> wrote:
>
>
>
>
>
>
>
> > Kelsey,
>
> > Thanks for the new page
> >http://onlinemathcircle.com/wiki/index.php?title=Irrational_numbers
>
> > I'd like you to add to it a proof of \sqrt(6) is irrational, and an
> > explanation of where the proof beaks down when you try to use the same
> > method to prove \sqrt(9) is irrational. This way you'll have looked at the
> > proof from more perspectives - which is good.
>
> > --James.
>
>
>
>
>
>
>
>
> > Kelsey,
>
> > Thanks for the new page
> >http://onlinemathcircle.com/wiki/index.php?title=Irrational_numbers
>
> > I'd like you to add to it a proof of \sqrt(6) is irrational, and an
> > explanation of where the proof beaks down when you try to use the same
> > method to prove \sqrt(9) is irrational. This way you'll have looked at the
> > proof from more perspectives - which is good.
>
> > --James.
>
> > On 1 July 2011 14:42, James Crook <james.k.cr...@gmail.com> wrote:
>
> >> Kelsey,
>
> >> I've added a tracking page to track progress on this course on the wiki:
> >>http://onlinemathcircle.com/wiki/index.php?title=Real_Analysis/Progress
>
> >> As you make progress I would like to see you add to the wiki, to make it
> >> clear which of these topics you've understood.
>
> >> Do you know the proof that square root of 2 is irrational? This will help
> >> us get through one of those topics.
>
> >> Also, do you have a copy of a book on real analysis such as Abbott to work
> >> from?
>
> >> This IS a first year university level course, and I am assuming you'd like
> >> to complete it in the study group time (i.e .by end August). If that is not
> >> important to you then we can go at whatever speed you like. If we do that
> >> you will get a flavor of what real analysis is about - though there's
> >> detail that you won't have 'got' by the end of August.
>
> >> --James.
>
>
> >> Kelsey,
>
> >> I've added a tracking page to track progress on this course on the wiki:
> >>http://onlinemathcircle.com/wiki/index.php?title=Real_Analysis/Progress
>
> >> As you make progress I would like to see you add to the wiki, to make it
> >> clear which of these topics you've understood.
>
> >> Do you know the proof that square root of 2 is irrational? This will help
> >> us get through one of those topics.
>
> >> Also, do you have a copy of a book on real analysis such as Abbott to work
> >> from?
>
> >> This IS a first year university level course, and I am assuming you'd like
> >> to complete it in the study group time (i.e .by end August). If that is not
> >> important to you then we can go at whatever speed you like. If we do that
> >> you will get a flavor of what real analysis is about - though there's
> >> detail that you won't have 'got' by the end of August.
>
> >> --James.
>
> >> On 27 June 2011 21:56, James Crook <james.k.cr...@gmail.com> wrote:
>
> >>> Kelsey,
>
> >>> Good, I'm satisfied you do understand the difference between uniform
> >>> continuity and continuity.
>
> >>> Notice that epsilon plays exactly the same role in each, it is only delta
> >>> that behaves differently. In one it is uniform for all x. It depends only
> >>> on epsilon. In the other it is a function of x as well as epsilon.
>
> >>> It may seem like the different kinds of continuity are a pernickety
> >>> detail, but it's incredibly important.
>
> >>> I've also added a formulation on this page:
>
> >>>http://onlinemathcircle.com/wiki/index.php?title=Quantifiers_don%27t_...
>
> >>> Kelsey,
>
> >>> Good, I'm satisfied you do understand the difference between uniform
> >>> continuity and continuity.
>
> >>> Notice that epsilon plays exactly the same role in each, it is only delta
> >>> that behaves differently. In one it is uniform for all x. It depends only
> >>> on epsilon. In the other it is a function of x as well as epsilon.
>
> >>> It may seem like the different kinds of continuity are a pernickety
> >>> detail, but it's incredibly important.
>
> >>> I've also added a formulation on this page:
>
James Crook
Jul 8, 2011, 5:24:15 PM7/8/11
to l2lea...@googlegroups.com
Sounds good.
I've put some questions about
on that page. When you're very familiar with what limits are you will be able to answer these - but until then they may be a bit puzzling. If you're unsure about how to answer any of them, you can say how far you got in an attempt, and I'll try and help.
I don't have 'Abbott' but if you have questions about exercises from it, you're welcome to ask about those exercises too.
--James.
James Crook
Jul 12, 2011, 6:05:45 PM7/12/11
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Kelsey,
That's good - I see you've had a go at some of these questions.
Is it helpful to have hints for some/all of these, or to give worked answers? (for any worked answer I will add another question to make up)
--James.
James Crook
Jul 23, 2011, 6:49:31 PM7/23/11
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Kelsey,
I've seen some more content on real analysis from you on the wiki, so I've added some more questions to go with it...
http://onlinemathcircle.com/wiki/index.php?title=Closed_sets--James.
Shri Ganeshram
Jul 23, 2011, 7:01:55 PM7/23/11
to l2lea...@googlegroups.com
Hey guys,
I've been slacking--though, I have a number of problems and solutions TeX'd up from various resources for a class I'm giving at AwesomeMath these next 3 weeks. I think these will be useful to import (depending on copyright) to the OMC wiki. If not, we'll figure out a way to make them materials for an OMC class.
--Shri
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