Another math problem! (Not HW)

[Kaylia]

Are you saying you want to start learning mathematics and you just need to know where to start/what direction to go? If that's what you're asking we could easily help you with that. But I don't want to post a bunch of stuff if that's not what you're saying.

I would take a bet and say it's impossible to learn mathematics on your own without exams and homeworks to motivate you.

It was possible back then, but the internet and video games weakened our poor mind and make us procrastinate too much.

[Woozie]

That is what I'm saying. I wanna be able to know what I'm looking at when I see a bunch of math symbols in front of me (namely Physics equations). I did buy a book on Calculus not too terribly long ago. I got about 1/3 of the way through it and promptly misplaced it when I moved. I'm not sure how much of that book I managed to retain... High School math was so long ago (9 years >.<) that I'm probably gonna have to relearn a lot of the Algebra I once knew.

Feel free to take it to PMs so as not to clutter this thread any further.

Edit: Have to get ready for work now, so I can't respond to any further posts/messages till tonight.

To be honest, you'll never understand most physics equations put in front of you. My own professors don't understand physics equations when they're from a field that they don't study. I mean, they understand the basics of every field, but if I show quantum field theory to solid state physicist, they'll only understand the parts of it that actually apply to solid state physics. If I go into theoretical physics, I'll probably never understand most of the equations specific to string theory because I don't plan to study that specific theory. I'm just telling you this so that you don't get an unrealistic idea that you'll be able to understand most equations that you see. Physicists specialize. I guess the same goes for math, but I don't notice it as much among the math professors. I can go to someone who studies analysis and ask a question about group theory, graph theory, or number theory, and they can usually answer it, though every once in a while I do get "Maybe you should go see professor [name]".

Either find your calculus book or get a new one. That's going to be the first step to understanding anything in math or physics. My personal favorite calculus book is the one by James Stewart.

If you get one of the thick calc books that goes through differentiation/integration/infinite series/three dimensional calculus/vector analysis, then you'll be done with calc 1, 2, and three when you finish the book.

After studying calculus, you'll be far enough along to study calculus-based physics, if you want to get into physics as well. I prefer Tiplers book

Amazon.com: Physics for Scientists and Engineers: Extended Version (9780716743897): Paul A. Tipler, Gene Mosca: Books

But Giancoli also makes a really good calc-based physics book. What you want is one of the big introductory books. It will have mechanics, fluid mechanics, thermodynamics, electricity and magnetism, and optics in it (and in that order). These are the freshman physics books.

You'll actually be ready for this physics book as soon as you know how to differentiate and integrate (so when you're about a quarter or a third through the calculus book, assuming you have a book with calc 1, 2, and 3 in it).

After freshman physics, you'll be ready for a book on modern physics. I don't have any specific book to recommend here.

Before you can go any further in physics, you will need to have finished calculus 3. It really helps if you've also studied ordinary differential equations. A quick way to get all the math you'll need (while ignoring any math you don't need specifically for physics) is if you get a book on advanced engineering mathematics or mathematical methods for the physical sciences. But if you really like the math, you'll want to get specific books on specific subjects. By the time you get to this level you'll probably have a good idea where to go depending on what you've studied. I'd recommend after going through differential equations that you then go through linear algebra. After linear algebra, you should study junior/senior level classical mechanics, electricity and magnetism, and quantum mechanics. By the time you get this far, you'll know where to find these books. When you get this far (it will take a while), you can just come back and we'll tell you where to go from there

If you're more into the math than the physics, you'll want to do abstract algebra, introductory analysis, and introductory topology after linear algebra. After this, you have so many options depending on what you like. You could go to graduate level analysis, more advanced topics in abstract algebra, more advanced levels of topology, differential geometry, number theory, etc. This is usually where mathematicians branch off and start studying more of what they like instead of doing a little bit of everything. So this would be the point where you'd stop understanding math from other fields except your own (so yeah, the same situation as with physics, but like I said, I don't notice it quite as much).

[Kaylia]

I think picking up a real number analysis book might be a nice place to start. It's basic algebra most of the time, and it really helps understanding the foundation of mathematics. Maybe someone who is new to mathematics wouldn't see the point behind certain proof though.

[joft]

I don't understand why lipschitz is more general, it should be the strictest, (> should be <)? I might be wrong here, but the way I see it

Lipschitz: College definition with derivative (doesn't work on fractal, cantor and other shitty function invented by evil mathematician)
Absolute: delta/epsilon on a f(x) function
Uniform: balls in a space with metric
topology: use of open and convergence of a set on another.

sorry I should have been more explicit. by > I meant "is a stronger condition"

the epsilon-delta definition is what I meant by "topological," because they're equivalent (assuming the topology is the one where open balls are defined by the metric). specifically, this type of continuity is defined pointwise-- a function is continuous at a point x if, whenever y is close enough to x, f(y) will be close to f(x). and a function is continuous on a set if it is continuous at each point in the set.

uniform continuity is a similar epsilon/delta definition but it's not defined at a specific point. in other words, you can't say a function is uniformly continuous at a point. the definition is like this: f is uniformly continuous on a set D if, whenever x and y are any two points in D sufficiently close to each other, f(x) and f(y) will also be close to each other. the key difference is that you aren't fixing one point and restricting closely to that one, you're insisting that the same amount of closeness (the same delta) will guarantee f(x) and f(y) will be close for ALL x and y that are within delta of each other. the previous type of continuity only requires that you can find a given delta that works for one point.
to put it in one sentence: for this one, delta depends ONLY on epsilon, it doesn't depend on where you are in the domain space, but for normal continuity delta can depend both on epsilon and where you are in the domain space.

example: the function f(x) = 1/x is continuous on (0,1]. but it is not uniformly continuous on this domain because it behaves badly when x approaches 0. so when x is closer to 1, it's very easy to guarantee f(x) and f(y) will be close if y is close to x. but when x is closer to 0, you will need y to be much closer to x in order to guarantee the same closeness of f(x) and f(y).

absolute continuity is an even stronger condition, and the definition is a bitch. it involves series. it goes like this: you give me epsilon. I can find a delta such that, if x_i and y_i, for i=1,2,...infinity is any infinite sequence of points such that the intervals [x_i,y_i] are all disjoint AND the infinite sum of all |x_i - y_i| is finite and less than delta then the infinite sum of all |f(x_i) - f(y_i)| will be less than epsilon. uniform continuity is the case where the sum only has 1 term.

Lipschitz continuity is the strongest of the 4 types I've mentioned, and the definition is easy: there exists a constant L such that |f(x) - f(y)| <= L*|x-y| for all x and y in the domain.

[joft]

Anyone interested in higher math would do well to check this book out:

Amazon.com: The Princeton Companion to Mathematics (9780691118802): Timothy Gowers, June Barrow-Green, Imre Leader: Books

It's fantastic.

[Sylvrdragon]

What got me interested in Physics in the first place was when I read Einstein's "Relativity" and then Hawking's "A Brief History of Time". I understand how incredibly dumbed down these works are (as they are meant to be), but it went over so many things that were immensely interesting to me. At the same time, though, I realize that Theoretical Physics isn't entirely practical. You aren't likely to find someone who is willing to pay you to do it. You are dependent on book sales, lol. As such, I'd like to find something (in the course of studying) that is both practical and interesting.

I consider myself an intelligent person, so I don't think there would be many fields too far over my head to undertake. I just find myself reading about things and having ideas, and then despairing that I don't have enough math to even know if my ideas are idiotic or not. If there was still money to be made off of short stories, then I might have become a Sci-fi writer by now.

[Kaylia]

I consider myself an intelligent person, so I don't think there would be many fields too far over my head to undertake. I just find myself reading about things and having ideas,

I can guarantee you the ride won't be as soft as you expect it to be.

At the same time, though, I realize that Theoretical Physics isn't entirely practical.

Unless you get a phd, then yes, it's mostly worthless.

But really, it's only worthless because the devil created engineering. You take easier classes, and they give you a pretty titles that can give you better jobs. Fucking unfair.

[joft]

phds are mostly worthless as well

90%+ of the quantitative jobs only require master's level training at most. beyond that is just preparation for academic research which almost no phds actually go on to do.

[Trajan]

I would take a bet and say it's impossible to learn mathematics on your own without exams and homeworks to motivate you.

It was possible back then, but the internet and video games weakened our poor mind and make us procrastinate too much.

I really have to agree with this post here lol. I have a complex analysis book in my room that I keep saying I am going to study, I think I've touched it once. Working on single problems like putnam exam problems or other undergrad math exam problems or math olympiad problems I do actually like to work on in my spare time, but starting an x hundred page math book is just too much, imo.

[Max™]

I am learning mathematics on my own.

Between various books I found while looking for stuff that Woozie has suggested to others, random wiki trawling, and Wolfram MathWorld: The Web's Most Extensive Mathematics Resource which is an awesome site...

Well, my eyes hurt, but it's quite possible if you're INTENSELY motivated.

I feel I need to be far better at math than I am, because as familiar as the math used in physics is... I am not as comfortable with pure mathematics, which sucks.

Incidentally, I was thinking you should apply curvilinear coordinates with a metric set such that the length from (0,0) to (1,1) is 1.

They are important for General Relativity, and if he was a physics major he'd know all about them.

[Woozie]

They are important for General Relativity, and if he was a physics major he'd know all about them.

Well yeah, but I think Julian is in an undergrad level class, so it doesn't seem fair that he'd have a problem that needed either a change in metric space or a change in geometry. I know it wasn't worth credit or anything (so it's not like he's getting an F for not solving it), but still...

[Woozie]

But really, it's only worthless because the devil created engineering. You take easier classes, and they give you a pretty titles that can give you better jobs. Fucking unfair.

I've always hated this. I joke about this a lot with engineers. They're really nothing more than applied physicist, yet they can go to school for four years, then easily get a job and make tons of money, whereas we have to go to school for 8~10 if we want to make money like them D:

[Khamsin]

I've always hated this. I joke about this a lot with engineers. They're really nothing more than applied physicist, yet they can go to school for four years, then easily get a job and make tons of money, whereas we have to go to school for 8~10 if we want to make money like them D:

It's because we actually do something! j/k, but really it's all about business. You can't sell physics equations, so naturally the guys who provide a service (namely, something you can charge clients for) are getting the money.

And along with my license, I get a shiny stamp with a seal and everything that says you can sue me if you get hurt because of my design.

[Kaylia]

I've always hated this. I joke about this a lot with engineers. They're really nothing more than applied physicist, yet they can go to school for four years, then easily get a job and make tons of money, whereas we have to go to school for 8~10 if we want to make money like them D:

Physics and maths make me hate everyone.

We had to take 3rd years engineering classes (electronics II) last semester. The six of us finished with B+ and above (average of the classes was B-) . It was my first time touching a breadboard or a diode, or using any programs they use to simulate stuff (pspice). Hell...I actually had to wikipedia "breadboard" during my first lab. I understand it might not be the hardest class they have..but still.

I know I'm generalizing, because the good engineer ARE good (and definitively better than I would if I working in the domain), but the whole thing still feel unfair. I don't care much though, I knew this before comming back to school in physics.

Well yeah, but I think Julian is in an undergrad level class, so it doesn't seem fair that he'd have a problem that needed either a change in metric space or a change in geometry. I know it wasn't worth credit or anything (so it's not like he's getting an F for not solving it), but still...

It wouldn't be a bad idea to bring your students' attention on the fact that metric, space, and operations can be altered, and lead to different (and sometime useful) mathematics.

If it was a normal question, I would agree with you, but bonus questions are the mindfuck that make curious people want to learn more.

Personally, I wish someone told me I could redefine addition and multiplication a long time ago. It's something I understood relatively recently, or at least, a long time after I was introduced to complex number for exemple (seriously...fuck sqrt(-1), the formal definition is so much better)


[edit[
Fixed a line that was murdered

[Ramor]

seriously, you're all thinking way too hard, the portals is the most correct answer for this problem

also, throughout my combing through the various responses given, did the TA ever give an official ruling?

[Kaylia]

sorry I should have been more explicit. by > I meant "is a stronger condition"

the epsilon-delta definition is what I meant by "topological," because they're equivalent (assuming the topology is the one where open balls are defined by the metric). specifically, this type of continuity is defined pointwise-- a function is continuous at a point x if, whenever y is close enough to x, f(y) will be close to f(x). and a function is continuous on a set if it is continuous at each point in the set.

uniform continuity is a similar epsilon/delta definition but it's not defined at a specific point. in other words, you can't say a function is uniformly continuous at a point. the definition is like this: f is uniformly continuous on a set D if, whenever x and y are any two points in D sufficiently close to each other, f(x) and f(y) will also be close to each other. the key difference is that you aren't fixing one point and restricting closely to that one, you're insisting that the same amount of closeness (the same delta) will guarantee f(x) and f(y) will be close for ALL x and y that are within delta of each other. the previous type of continuity only requires that you can find a given delta that works for one point.
to put it in one sentence: for this one, delta depends ONLY on epsilon, it doesn't depend on where you are in the domain space, but for normal continuity delta can depend both on epsilon and where you are in the domain space.

example: the function f(x) = 1/x is continuous on (0,1]. but it is not uniformly continuous on this domain because it behaves badly when x approaches 0. so when x is closer to 1, it's very easy to guarantee f(x) and f(y) will be close if y is close to x. but when x is closer to 0, you will need y to be much closer to x in order to guarantee the same closeness of f(x) and f(y).

absolute continuity is an even stronger condition, and the definition is a bitch. it involves series. it goes like this: you give me epsilon. I can find a delta such that, if x_i and y_i, for i=1,2,...infinity is any infinite sequence of points such that the intervals [x_i,y_i] are all disjoint AND the infinite sum of all |x_i - y_i| is finite and less than delta then the infinite sum of all |f(x_i) - f(y_i)| will be less than epsilon. uniform continuity is the case where the sum only has 1 term.

Lipschitz continuity is the strongest of the 4 types I've mentioned, and the definition is easy: there exists a constant L such that |f(x) - f(y)| <= L*|x-y| for all x and y in the domain.

Sorry, I skipped your post last night because I was too tired, and topology is at the limit of my mathematicals knowledge.

After reading my old notes, I realize that your definitions are correct. I confused uniform continuity with the generalization of absolute continuity to every spaces. I dont think I ever used it in a demonstration either. It's kinda implied in Lebesgue integral, but we didn't use formal definition for this.

[Max™]

I meant the teacher was a physics head, and I was reading the thread thinking the whole time "why not just convert the metric after converting to a curvilinear coordinate system", so it seems like an obvious answer to me.

That, portals, or draw the diagonal, measure the length of a line segment, then note that you added a t dimension to the sheet, and demonstrate by curving a sheet of graph paper until it's the same length as one segment.

Tell him "hey, this is valid, it's physics!"

[Woozie]

Since we have a bunch of math people here, I may as well ask this here (this isn't homework)

How exactly do I prove that every cyclic group of infinite order is isomorphic to the group of integers under addition? I know for sure that this is true but I can't find any proofs online. It's been a while since I've done isomorphisms D:

If anyone knows a site that proves it or feels like typing up a proof, it would really help. I need to use this fact for a project I'm working on, and even though I'm 100% certain it's true, I'd just feel more comfortable if I see the proof.

[Max™]

Cyclic Group -- from Wolfram MathWorld

Also: http://mathworld.wolfram.com/Kroneck...onTheorem.html

I remember these from learning about Abelian groups.


Any finite abelian group can be written as isomorphic to Z or to a sum of cyclic groups.

[Woozie]

That site doesn't have a proof.

Edit: The site does mention that any abelian group can be expressed in terms of Zn groups, but they're clearly referencing the fundemental theorem of finite abelian groups (so I can't just apply that theorem to an infinite group). What I'm looking for is proof that any infinite cyclic group is isomorphic to Z.