Self-Study Group: Join Now Before Semester Ends!

[PhysicalAnomaly]

First, I would like to point out that I will be quite preoccupied until the 20th of June. Winter break doesn't start till then. I can however still do this though I might be giving fewer lectures than receiving. Also, I haven't had a chance to check your proposed website out but I'm not sure how we'd do this in real time if that's what you're planning since I'm probably over half a day ahead.

I have already been through a course that does grad, div, curl, line and surface integrals and Stokes, Gauss, etc. This wasn't done rigorously but I doubt we'd be doing that here either. I was thinking (hoping?) that we'd all have at least done these topics since they constitute the very basics of undergrad maths, usually taken in 1st or 2nd year. (*sigh*) It would probably be counterproductive to go through all of it in detail, especially since each of you have done some (the ones that the others haven't done XD). We could do a quick run-through...

I was thinking that we'd do topics such as:

Jordan normal form, nilpotent operators, tensors, Hermitian matrices
DE's, finite difference and shooting methods, Laplace transforms, numerical methods in DE's such as WKB and Sturm-Liouville (But no advanced stuff like Sobolev spaces)
Laguerre, Legendre, Airy functions (used in QM)
Brownian motion, random walks, Markov processes, Monte Carlo methods (useful for stats and probability, statistical physics and QM)
Langrangian formulation of mechanics

 

[Jacobpm64]

you have a good list of stuff there anomaly :)

 

[PhysicalAnomaly]

Hmmm... I see one disadvantage of the chatroom - you can't record sessions. It would be nice to be able to in case a member misses a session. Also, it would be nice for revision.

 

[Jacobpm64]

Anomaly, I just typed random math stuff in the chat box in MathIM and copy/pasted it into a microsoft word document. Everything showed up well.

Works fine to me.

 

[millitiz]

I was thinking that we'd do topics such as:

Jordan normal form, nilpotent operators, tensors, Hermitian matrices
DE's, finite difference and shooting methods, Laplace transforms, numerical methods in DE's such as WKB and Sturm-Liouville (But no advanced stuff like Sobolev spaces)
Laguerre, Legendre, Airy functions (used in QM)
Brownian motion, random walks, Markov processes, Monte Carlo methods (useful for stats and probability, statistical physics and QM)
Langrangian formulation of mechanics

Hmm, now I feel very ashamed... I only know half of the listed items, and the other half of them, I either have no idea, or only remotely know/hear of them...
The worst part is that I am already a junior/senior in college, and also major in math/physics...

And if we are talking about proving them... That'll be FASCINATING! (and my quick estimation is that it could take at least an year to thoroughly go over them.)
For instance, there are linear algebra/real analysis, Differential Geometry, ODE, PDE, (Topology is probably needed in some cases) etc. They all worth at least a semester of studying. But even just thinking of it makes me exciting! Proofs, proofs, and more proofs!

Oh well, I doubt if that is ever going to be the case anyway...

 

[PhysicalAnomaly]

I don't know about them but I'll be proving whatever I do (of course, I might have to hand-wave a bit). And yeah, I'm only in my second year. =P But I don't know more than a brief definition of them. Join us. I'll only be doing this very casually - as I said, it will be while my semester is on. In fact, I think we should just start now and do it casually as a long-term hobby. I just don't see how we'll learn stuff well if we rush.

@jacob: Do the symbols appear on word though?

 

[Jacobpm64]

the symbols sure do appear on word.. because MathIM uses the latex code you type.. and makes it into images. Therefore, when you copy and paste the text from the chat window into microsoft word.. You're pasting words.. with images (the math is just like inserting a photo into a word document).

 

[millitiz]

I don't know about them but I'll be proving whatever I do (of course, I might have to hand-wave a bit). And yeah, I'm only in my second year. =P But I don't know more than a brief definition of them. Join us. I'll only be doing this very casually - as I said, it will be while my semester is on. In fact, I think we should just start now and do it casually as a long-term hobby. I just don't see how we'll learn stuff well if we rush.

@jacob: Do the symbols appear on word though?

Hmm, you are confusing me further. I thought this self studying is more on the applicational, engineer-physics type of math (and that was the reason I decided to quit, why, I already know them).
And your listed topics, although some of them are well related, some of them could be a whole field in its own right. And I don't see how it could be done in a summer, if you are thinking of anything of mathematically rigorous.
Let's just pick tensor for an example.
Tensor is in fact, related to differential geometry, or greometry in general.
So to learn tensor, one also need to learn forms. And I think it is also related to Cohomology (I think this is the word?). And since it is "differential" geometry, one also need to learn differentiation, Rienman integration, (possibly Lesbegue integration?). Of course we know how to do differentiation, integration, and what not. But what is differentiation? In what condition can we do differentiation or integration (in fact, this is where lesbegue comes in, if you know it ;D)? And also, one need to know about space, sets, open, closed, compact, etc topology.
Yea, so I am currently learning some differential geom now in an analysis class(?). And use it to prove Stoke's theorem. Needless to say, it is fascinating!
Nonetheless, I can still see that some of them are more narrowed than others, and could be done quickly (or maybe I am just not knowledgeable in those fields?)

But to be honest, except those math nerds like us, I doubt anyone is interested in step into this "mine field."

But then again, we can always make another subgroup. So it kinda depends on how people think about it.

 

[PhysicalAnomaly]

I was thinking that it would be on during semesters as well. Also, the proofs will be presented but not everyone has to understand them. The concepts themselves can be understood (intuitively anyway) without proofs. A lot of those concepts can be learned on their own (assuming basic undergrad applied maths) in the manner that physicists and statisticians do.

And no, the most general form of a tensor is not due to differential geometry. It is an algebraic object along with a multilinear mapping. But we can of course avoid showering applied peeps with 'abstract nonsense' as they'd like to call it and only learn specific tensors as used in civil engineering, relativity, differential geometry, etc. In many ways, these are the things that would be of most benefit to pure mathematicians who don't get a chance to see it in their course (because they're too busy learning about elliptic curves and homologies).

 

[millitiz]

I was thinking that it would be on during semesters as well. Also, the proofs will be presented but not everyone has to understand them. The concepts themselves can be understood (intuitively anyway) without proofs. A lot of those concepts can be learned on their own (assuming basic undergrad applied maths) in the manner that physicists and statisticians do.

Yes, I understand. I was a physics student originally, I totally understand that concepts could be learned and "proved" by hand waving ;).

And no, the most general form of a tensor is not due to differential geometry. It is an algebraic object along with a multilinear mapping. But we can of course avoid showering applied peeps with 'abstract nonsense' as they'd like to call it and only learn specific tensors as used in civil engineering, relativity, differential geometry, etc. In many ways, these are the things that would be of most benefit to pure mathematicians who don't get a chance to see it in their course (because they're too busy learning about elliptic curves and homologies).

Thanks for teaching. As I said, I was a physics students, and didn't think of taking any advance math course until this year. So yea, I guess I don't really have that deep knowledge in math (yet, I hope). But I am going to take modern algebra the coming school year, so I guess I'll learn it then?
And yes, the first(and probably the only time?) I know the tensor is within the context of relativity, where tensor is a map mapping vectors and forms, which then I kinda assume that it is related to Diff Geom. Well, where are the others?

 

[PhysicalAnomaly]

Don't worry, I'm only a second year student and only started on any serious pure maths late last year and early this year. You'll catch up in no time. You probably won't be studying tensors of algebras in a first course in algebra though. The reason I got here so fast is that I didn't take a first course... nor a second.

Yeah, they have k-tensors. I think relativity got their tensors from diff geometry. In civil engineering, they have stress tensors. That probably came straight out of multilinear algebra.

The rest of them are probably busy studying. I'm trying to do 10 questions a day. Which doesn't sound like much but some questions take a few hours of thinking to complete. What about you?

 

[djeitnstine]

Yes, studying for finals has been time consuming. I have the last four between today and tomorrow.

I'll probably end up doing my own self-studying, or perhaps millitiz can help me (with all the engineering/physics application type math) since you guys are on your way to pure math,

However, if some of you are still up for Fluid Dynamics, I'll definitely get involved in that.

 

[millitiz]

10 questions per days! Man, that is a TON of problems. Sometimes I couldn't even figure out A question for hours (I think the worst ones took me for DAYS) :).
Well, actually, I am doing some reading outside classes, but not really a lot (at least not 10 questions per day!), and btw, final is coming.
You see, 4 problems per week is a hell load of homework for our school (both physics and math). And they could easily take four five pages. So 10 problems per DAY is amazing for me :)
(How many trees do you kill per day! XP)

 

[millitiz]

Yes, studying for finals has been time consuming. I have the last four between today and tomorrow.

I'll probably end up doing my own self-studying, or perhaps millitiz can help me (with all the engineering/physics application type math) since you guys are on your way to pure math,

However, if some of you are still up for Fluid Dynamics, I'll definitely get involved in that.

I think Anomaly is suggesting some of the applicational aspects of those topics. And basically no proof involved. Although I do need to admit that the list s/he gave was quite different from what you/I expect...

 

[PhysicalAnomaly]

10 questions per days! Man, that is a TON of problems. Sometimes I couldn't even figure out A question for hours (I think the worst ones took me for DAYS) :).
Well, actually, I am doing some reading outside classes, but not really a lot (at least not 10 questions per day!), and btw, final is coming.
You see, 4 problems per week is a hell load of homework for our school (both physics and math). And they could easily take four five pages. So 10 problems per DAY is amazing for me :)
(How many trees do you kill per day! XP)

I don't always manage ten. And yeah, I only do one or two problems that take me hours/days to do. The rest are half hour/15 min problems. Get a book with hard questions and start cracking? And yeah... erm... I tend to write concisely and squeeze things a bit. I don't write cursive so it's still quite legible. I would like to say that it's elegant and short but I end up cancelling quite a bit and take the long way sometimes. It's only a few pages a day though. I probably kill a small tree per semester (more worried about the ink I use - my father is left wondering why I'm buying pens every other week). ;)

@dj-something: Don't be put off. As millitiz said, it'll be mostly the applied side of it. Even I'm not prepared to study DE's in depth. We'll go over grad, div, curl, Gauss, Kelvin-Stokes, Green first. All the topics I've mentioned are routinely studied by 3rd and 4th year students in engineering, physics and maths. I don't expect everyone to have been through real analysis (though you do have my scorn if you haven't =P).

PS Oh and I've been neglecting physics. It's terrible but maths has consumed my soul. I'm hoping that my maths experience will negate any disadvantage I'll have from neglecting physics. XD

 

[Jacobpm64]

Hey guys, I'm just letting y'all know that I'm working these last two weeks to finish this semester off the best that I can.. My last day of school is May 8th. Then, I'll be ready to do whatever.

 

[Tickitata]

I'm a second year pure mathematics major

If anyone would like to set up an Algebra, Analysis, Topology, or Int. E&M summer study group, let me know. I'd love to set up something like this, it sounds like a really great idea

 

[naele]

Tickitata, put me down as interested for either of those

 

[PhysicalAnomaly]

Is this still going ahead?

 

[djeitnstine]

I don't know seems as if it died. I haven't heard from anyone in a while so I started to study PDE's and continuum mechanics.

Honestly I've sort of strayed away since everyone wanted proof based material. But I can be flexible.

What is everyone thinking?

 

[PhysicalAnomaly]

I'm thinking that you should start on multivariable calculus. It's a good area where derivations and proofs go together very well. Start with vector fields and then move on to vector bundles and tensors perhaps? That's pretty important for physicists and some engineers. Or you could go the divgradcurl and integral theorems route. Maybe even differential forms. I'm at the climax of my semester so you'll have to take the lead.

 

[PhysicalAnomaly]

And yes, those are applied topics. Almost all the topics I have mentioned are applied maths topics. If you don't believe me, look up: Boas, Stroud, Kreyzieg, Arfken...

 

[naele]

I'm gearing up for a linear algebra study group on another forum that would start in the 3rd week of June. I've already got a problem set for some set theory concepts put together and I will be starting some linear algebra problem sets soon. If there's interest I can simultaneously do it here. Here's the preliminary curriculum
1. Systems of linear equations
- Gaussian, Gauss-Jordan elimination, solution sets, general + particular solution, etc.
- Matrix operations, elementary row operations, etc
2. Vector spaces
- Subspaces, spanning sets, independence, basis, dimension, etc
3. Linear Maps
- Null spaces, range, matrix representation, change of basis, projection, etc
4. Determinants
- n-dimensional determinants, properties, existence and uniqueness, laplace expansion, Cramer's rule, etc
5. Eigenvalues and Eigenvectors
- Eigenvalue problem, eigenvectors, diagonalization, similarity, nilpotence, etc
6. Inner product spaces
- Inner products, norms, orthogonality, adjoints and self-adjoints, etc
7. Canonical forms
- Jordan form, generalized eigenvectors, etc

 

[djeitnstine]

And yes, those are applied topics. Almost all the topics I have mentioned are applied maths topics. If you don't believe me, look up: Boas, Stroud, Kreyzieg, Arfken...

Ok sounds good. I'll be available MAX 2x a week, so that should be good enough since we're both busy.

Umm I'll pm you with the information so that we can set this up. We need an easily accessible book to use and decide upon a syllabus - which you seem to have mostly sorted out already. I have a great deal of e-books in my stash I can sieve through in which I can share.

 

[PhysicalAnomaly]

I'm gearing up for a linear algebra study group on another forum that would start in the 3rd week of June. I've already got a problem set for some set theory concepts put together and I will be starting some linear algebra problem sets soon. If there's interest I can simultaneously do it here. Here's the preliminary curriculum
1. Systems of linear equations
- Gaussian, Gauss-Jordan elimination, solution sets, general + particular solution, etc.
- Matrix operations, elementary row operations, etc
2. Vector spaces
- Subspaces, spanning sets, independence, basis, dimension, etc
3. Linear Maps
- Null spaces, range, matrix representation, change of basis, projection, etc
4. Determinants
- n-dimensional determinants, properties, existence and uniqueness, laplace expansion, Cramer's rule, etc
5. Eigenvalues and Eigenvectors
- Eigenvalue problem, eigenvectors, diagonalization, similarity, nilpotence, etc
6. Inner product spaces
- Inner products, norms, orthogonality, adjoints and self-adjoints, etc
7. Canonical forms
- Jordan form, generalized eigenvectors, etc

I should hope that anyone who's doing the study group with us has at least a basic understanding of these topics. Anyone who isn't is not really prepared to study anything else pure or applied.

And yes, do pm me. We can adjust the syllabus as we go along according to interest. Likewise with the tone. As for books, I was thinking that we could all use different books (whatever we have access to) and then we can share stuff that one or the other doesn't have. But yeah, your way is probably better.

 

[djeitnstine]

As for books, I was thinking that we could all use different books (whatever we have access to) and then we can share stuff that one or the other doesn't have. But yeah, your way is probably better.

Ok cool, I can get stuff cranking tomorrow since I get a day off - finally. I just really wanted everyone to have the same problem sets at least. So if there's any difficulty or something it won't be hard to help the other.

 

[PhysicalAnomaly]

What's the first topic?

 

[Jacobpm64]

Yeah, I'm here... Just PM me when y'all get something together.. and I'll tag along... I'm on summer break now for 3 months. :-)

 

[Winter Flower]

Hi. If you guys are still looking for more members, I am somewhat interested in joining. I am going to be a senior (physics/astrophysics major), and am interested in math but have had some difficulties in my past classes. This sounds like it would be helpful and fun...

 

[Pyrrhus]

Batchelor is a classic in Fluid Dynamics. It's a good book.

 

[djeitnstine]

Hi. If you guys are still looking for more members, I am somewhat interested in joining. I am going to be a senior (physics/astrophysics major), and am interested in math but have had some difficulties in my past classes. This sounds like it would be helpful and fun...

Sure just give us some time to sort things out and i'll give you a message

 

[Winter Flower]

Sure just give us some time to sort things out and i'll give you a message

Cool. Thanks. :)