How much Maths does one need in Particle Physics?

[Lavabug]

I have the same question as the OP, but my situation is different: I'm a 3rd year student and I've finished all my math courses (linear algebra(weak on theory), year's worth of multivariable calculus, ODEs, PDEs, complex variables and integral transforms). Which means I have no option of taking more math in my 3rd and 4th years unless I spend my 4th year abroad and manage to take something.

My university used to offer an elective introductory course on tensors, integral equations and group theory with applications to physics in the 3rd year. I managed to get the notes on the subjects but they're pretty condensed/summarized. Can anyone recommend a good book on these subjects that's accessible for self-study/fun? Because it sounds like it would be extremely relevant to any sort of advanced physics in particular HEP.

Is rigor really important at this level? Or is it more important to get familiarized with applying these areas of math to physics than having an "epsilon-delta analysis" emphasis?

 

[FeDeX_LaTeX]

I apologise if i pulled a 'heart string'. What I meant was that throughout school I am constantly taught Mathematics that seems to have no practical use like Median, Prime Factors, HCF, LCM etc. Personally I love Algebra.

I'm assuming you mean GCSE Mathematics. Yes, your GCSE Maths years (14-16) will be incredibly boring. Start learning some AS topics; it couldn't hurt. You could do what I did and get your school to enter you for some AS modules early if you're confident enough to; that will be a starting point into calculus. One of the most intimidating things about mathematics can often be notation for some students.

Yes, there is one. It's called "The Road to Reality" by Roger Penrose. If you read it cover to cover, and understand everything in it, you will know all you need to know to be a great physicist.

Road to Reality is a great book, but it is no easy read. Despite what some reviews and the blurb may suggest, you do need to do a lot of background research when it comes to the harder chapters. I doubt anyone can use this book alone to teach themselves the mathematics of the book, particularly the manifolds chapters (and the Lie algebra chapter, I think?). Although this book is cheap, it certainly is not for the scientific layperson. I disagree with Penrose's view that you don't have to look at the equations in detail to understand what is going on. Simply put, if you don't have the sufficient mathematical background to learn some of these topics, you won't learn much from this book. I recommend that you make an attempt (because it can be a very rewarding read) but don't become discouraged if you make almost no progress with it. Don't be fooled by the cover; this book will have a lot of mathematics in it. There are exercises at the bottom of most pages for you to verify your learning, but they aren't easy.

I've canceled the order for the Quantum Physics book and I think I'm going to get that "Basic Mathematics" book by Serge Lang.

I think the best thing for you to do is actually just to get an AS textbook. You can even get them for free online these days from a quick Google search. Get the Edexcel ones; there's one for each of the 18 modules; C1-4, M1-5, FP1-3, D1-2, S1-4. Start working your way up through the core modules and do some of the mechanics and further pure ones (stats couldn't hurt too much either). I think the D1/D2 books are pretty bad though; they do teach the content, but they are terrible representations of Discrete Mathematics (the manner in which it is taught it notoriously dull). Get the newer textbooks (2008 onwards); they make the stuff easy to understand and have lots of exercises. Not only that, but the way in which these textbooks are laid out means your knowledge shouldn't atrophy, especially as you're going to be sitting those papers pretty soon anyway!

Also take a look at the NRICH website (nrich.maths.org). There are lots of problems on there for you to solve. Despite its colourful look, some of those problems are very tough; if you find a problem really easy, you can consider possible extensions of the problem and consider the notion of proof.

I also recommend taking some time out every now and then. Do you like problem-solving? Seeing as you're in the UK (like me) and at that age where you start to become 'better' at mathematics (for me, anyway), it wouldn't hurt to try experimenting with some of the mathematics you've learned already. You might thing that some of the GCSE stuff is completely pointless, but they are fundamental prerequisites for A-level topics (mostly -- there is some stuff in the GCSE spec that you'll never need for AS/A2/STEP, ever). For instance, at your age, I wondered why you could only find natural-number derivatives (1st, 2nd, 3rd, etc.), but with some experimentation for a day or two I found that you could extend it to other sets of numbers (e.g. it became possible to find the (3+2i)th derivative of some function). Experiment with what you have, and try to solve some problems (NRICH); it'll be very rewarding. Unless you hate maths with a passion.

I'm hearing a lot of things from people like "take precalc, trig, algebra II, (...)". In the UK you probably won't know what those topics entail, so I think it's easier if I just say which modules to look at: Pre-calc stuff is higher-end GCSE-level and covered in C1-C4; same for trig. I'm tempted to say not to waste too much time with C1 since half of it is GCSE stuff, but its foundations are important.

I was in your situation just two years ago; volunteer for masterclasses where they are offered, look at online lectures (KhanAcademy, PatrickJMT, Dr Chris Tisdell (sp?), and there's a guy that does lots of Topology stuff too, but look at that when you're in your AS year). I started off with Edugratis (whose website is sadly no longer working) who introduced me to calculus at 13-14. That led me to "Paul's Online Math Notes". Take your time and work through it slowly.

And don't neglect your other subjects!

If you need resources or have any queries, you can PM me if you wish. I'm 16 and doing my A-levels and (hopefully) can give you a bit of guidance.

 

[dalcde]

Don't really care about the rest of Physics besides the particle aspect, e.g.I couldn't give a damn about light refraction or thermal conduction.

I'm with you.

 

[MarcAlexander]

I'm assuming you mean GCSE Mathematics. Yes, your GCSE Maths years (14-16) will be incredibly boring. Start learning some AS topics; it couldn't hurt. You could do what I did and get your school to enter you for some AS modules early if you're confident enough to; that will be a starting point into calculus. One of the most intimidating things about mathematics can often be notation for some students.



Road to Reality is a great book, but it is no easy read. Despite what some reviews and the blurb may suggest, you do need to do a lot of background research when it comes to the harder chapters. I doubt anyone can use this book alone to teach themselves the mathematics of the book, particularly the manifolds chapters (and the Lie algebra chapter, I think?). Although this book is cheap, it certainly is not for the scientific layperson. I disagree with Penrose's view that you don't have to look at the equations in detail to understand what is going on. Simply put, if you don't have the sufficient mathematical background to learn some of these topics, you won't learn much from this book. I recommend that you make an attempt (because it can be a very rewarding read) but don't become discouraged if you make almost no progress with it. Don't be fooled by the cover; this book will have a lot of mathematics in it. There are exercises at the bottom of most pages for you to verify your learning, but they aren't easy.



I think the best thing for you to do is actually just to get an AS textbook. You can even get them for free online these days from a quick Google search. Get the Edexcel ones; there's one for each of the 18 modules; C1-4, M1-5, FP1-3, D1-2, S1-4. Start working your way up through the core modules and do some of the mechanics and further pure ones (stats couldn't hurt too much either). I think the D1/D2 books are pretty bad though; they do teach the content, but they are terrible representations of Discrete Mathematics (the manner in which it is taught it notoriously dull). Get the newer textbooks (2008 onwards); they make the stuff easy to understand and have lots of exercises. Not only that, but the way in which these textbooks are laid out means your knowledge shouldn't atrophy, especially as you're going to be sitting those papers pretty soon anyway!

Also take a look at the NRICH website (nrich.maths.org). There are lots of problems on there for you to solve. Despite its colourful look, some of those problems are very tough; if you find a problem really easy, you can consider possible extensions of the problem and consider the notion of proof.

I also recommend taking some time out every now and then. Do you like problem-solving? Seeing as you're in the UK (like me) and at that age where you start to become 'better' at mathematics (for me, anyway), it wouldn't hurt to try experimenting with some of the mathematics you've learned already. You might thing that some of the GCSE stuff is completely pointless, but they are fundamental prerequisites for A-level topics (mostly -- there is some stuff in the GCSE spec that you'll never need for AS/A2/STEP, ever). For instance, at your age, I wondered why you could only find natural-number derivatives (1st, 2nd, 3rd, etc.), but with some experimentation for a day or two I found that you could extend it to other sets of numbers (e.g. it became possible to find the (3+2i)th derivative of some function). Experiment with what you have, and try to solve some problems (NRICH); it'll be very rewarding. Unless you hate maths with a passion.

I'm hearing a lot of things from people like "take precalc, trig, algebra II, (...)". In the UK you probably won't know what those topics entail, so I think it's easier if I just say which modules to look at: Pre-calc stuff is higher-end GCSE-level and covered in C1-C4; same for trig. I'm tempted to say not to waste too much time with C1 since half of it is GCSE stuff, but its foundations are important.

I was in your situation just two years ago; volunteer for masterclasses where they are offered, look at online lectures (KhanAcademy, PatrickJMT, Dr Chris Tisdell (sp?), and there's a guy that does lots of Topology stuff too, but look at that when you're in your AS year). I started off with Edugratis (whose website is sadly no longer working) who introduced me to calculus at 13-14. That led me to "Paul's Online Math Notes". Take your time and work through it slowly.

And don't neglect your other subjects!

If you need resources or have any queries, you can PM me if you wish. I'm 16 and doing my A-levels and (hopefully) can give you a bit of guidance.

Thank you. Finally someone who realizes that things in America have different naming conventions as of that in the UK. I know how to do Trigonometry and I enjoy 'problem solving', I just dislike graphs very much as I have never really found a use for them besides the obvious uses. I shall pursue my research into these areas of Mathematics but mainly focus on current studies on it(GCSE ones). I'm going to go do my Chemistry homework, will take a few hours or so, I'm going to say: live long and prosper.

 

[FeDeX_LaTeX]

Thank you. Finally someone who realizes that things in America have different naming conventions as of that in the UK. I know how to do Trigonometry and I enjoy 'problem solving', I just dislike graphs very much as I have never really found a use for them besides the obvious uses. I shall pursue my research into these areas of Mathematics but mainly focus on current studies on it(GCSE ones). I'm going to go do my Chemistry homework, will take a few hours or so, I'm going to say: live long and prosper.

What has been your experience of trigonometry so far? If it's just the stuff in the GCSE syllabus, make sure you remember all of it for A-level. You'll need them for M1-5 and C2.

I'm not sure what you mean by 'graphs'. If you mean the standard y = mx + c and y = ax^2 + bx + c graphs they teach you at GCSE, there is a lot more to it than just that, especially in statistics (t-distributions, probability density function, continuous and cumulative distribution functions, etc. -- these are all S1-4 topics). What is it that you don't like about graphs? Is it the notation that is confusing ( i.e. f(x) )?

After studying C1 you will probably like graphs a bit more. Have you heard of limits?

 

[neutrino']

marcalexander. if you know the famous particle physicist Richard Feynman, he had alredy mastered calculus by the age of 15 which means that it is time for you to start enlighten yourself of calculus. He discuses in his autobiography that his mastery of calculus at young age helped him at undergrad school (at MIT), which is clear that his success in QED and other researchs arises from his mastery of calculus. He studied calculus using the book "Calculus for the practical man" buy it or google it for ... as FeDeX_ LaTeX said a quick googling ;).

your consideration for your future career is appreaciable.

 

[FeDeX_LaTeX]

Just out of interest -- what constitutes a 'mastery of calculus'?

 

[MarcAlexander]

What has been your experience of trigonometry so far? If it's just the stuff in the GCSE syllabus, make sure you remember all of it for A-level. You'll need them for M1-5 and C2.

I'm not sure what you mean by 'graphs'. If you mean the standard y = mx + c and y = ax^2 + bx + c graphs they teach you at GCSE, there is a lot more to it than just that, especially in statistics (t-distributions, probability density function, continuous and cumulative distribution functions, etc. -- these are all S1-4 topics). What is it that you don't like about graphs? Is it the notation that is confusing ( i.e. f(x) )?

After studying C1 you will probably like graphs a bit more. Have you heard of limits?

I mean bar charts and median etc. I love linear algebra. ;)
No, I have not heard of this 'limits'.

 

[MarcAlexander]

marcalexander. if you know the famous particle physicist Richard Feynman, he had alredy mastered calculus by the age of 15 which means that it is time for you to start enlighten yourself of calculus. He discuses in his autobiography that his mastery of calculus at young age helped him at undergrad school (at MIT), which is clear that his success in QED and other researchs arises from his mastery of calculus. He studied calculus using the book "Calculus for the practical man" buy it or google it for ... as FeDeX_ LaTeX said a quick googling ;).

your consideration for your future career is appreaciable.

A well spoken post. ;)

As soon as I have 'remembered' all of my previous Mathematical education(i.e.revised it), then I shall look into Calculus and maybe even start learning it. :)

 

[neutrino']

Just out of interest -- what constitutes a 'mastery of calculus'?

understanding ("mastery") of differential and integral calculus. and then partial differentiation and so forth. also mastering multivariable differentiation.

FeDeX_ LaTeX what r u trying to pull?


and marc, I should have mentioned that mastering precalulus (having a solid background on trig, matrix...) is important. Take a look at this book:
Precalculus, Larson
it is a very nice book with practical application of knowledge of each chapter.

 

[FeDeX_LaTeX]

I wasn't trying to "pull" anything; merely trying to see what it takes to 'master' calculus. Given the amount of different topics in calculus that does seem like a difficult task.

C1 will teach you how to differentiate and integrate a basic function in the form of kxⁿ.
C2 and M2 will open you up to applications of differentiation.
C3 will teach you how to differentiate almost every common function.
C4 will teach you methods of integration and implicit differentiation which can then be applied in M3.

 

[neutrino']

I wasn't trying to "pull" anything; merely trying to see what it takes to 'master' calculus. Given the amount of different topics in calculus that does seem like a difficult task.

C1 will teach you how to differentiate and integrate a basic function in the form of kxⁿ.
C2 and M2 will open you up to applications of differentiation.
C3 will teach you how to differentiate almost every common function.
C4 will teach you methods of integration and implicit differentiation which can then be applied in M3.

Excuse me fedex. I shouldn't have said that. Since I don't live in the UK, I want to know if you are given Calculus in the age 14-17?

 

[MarcAlexander]

Excuse me fedex. I shouldn't have said that. Since I don't live in the UK, I want to know if you are given Calculus in the age 14-17?

Calculus is not available until you reach A-level/University, I think. I do not possesses access to learning Calculus at school, yet.

 

[neutrino']

so A-level means university? i have uk physics books. so what does O- level mean?

 

[GregJ]

GCSEs followed by O-Levels. After those are A-Levels (can be thought of as university entrance level). Although I started school in the UK, I did not finish so I am not 100% sure.

 

[neutrino']

GCSEs followed by O-Levels. After those are A-Levels (can be thought of as university entrance level). Although I started school in the UK, I did not finish so I am not 100% sure.

ok. but can somebody list o , a ... levels in accordance with age? eg age X - _ level

lets finish it here since we are outta topic
thx in advance.

 

[MarcAlexander]

ok. but can somebody list o , a ... levels in accordance with age? eg age X - _ level

lets finish it here since we are outta topic
thx in advance.

GCSE=new version of O-level(14-16)
A-level=College/6th Form(16-18)
...And then you do degrees at University.

 

[neutrino']

thx marc

 

[MarcAlexander]

thx marc

Your welcome. ;)

 

[FeDeX_LaTeX]

Your GCSE years (or what would have been called O-levels a long time ago) are done over two years in schools; Year 10 (aged 14-15) and Year 11 (aged 15-16). Your A-levels consist of an AS year and an A2 year; Year 12 (aged 16-17) and Year 13 (aged 17-18).

A student studying AS Mathematics only will do 3 modules; on the Edexcel exam board, these are C1, C2, and anyone from D1, S1 or M1. A student studying A-level Maths over two years does C1, C2, and anyone from D1, S1 or M1 in Year 12, and C3, C4, and another applied module. There are limitations as to which applied module combination you can do. For example, you can do M1 + M2, S1 + S2, S1 + M1, and D1 + D2, but not S1 + D2, or S2 + M2.

A student studying A-level Further Maths will do those 6 modules listed above (C1-4 + 2 applied) in Year 12, and then another 6 in Year 13 (FP1, either one of FP2 or FP3, and 4 applied units / 3 if you did FP1+FP2+FP3).

Here's a list of what's taught in each module for Edexcel in the UK.

http://www.edexcel.com/migrationdocuments/GCE New GCE/UA024850 GCE in Mathematics issue 2 180510.pdf

There is a third qualification available called Further Maths (Additional) which requires that you do all 18 modules (C1-4, FP1-3, D1-2, S1-4, M1-5).

The grading works as follows; to gain an A in Maths you need 480/600 total over the 6 modules. To gain the A*, you need 480/600 and an average of 90 in C3 and C4 combined. For Further Maths, you need 480/600 for the A, but an average of 90 across your 3 best modules for the A*.

At the beginning of Year 13, you apply to universities with your AS grades, and sometimes they ask you for an interview and possibly give you an offer, usually based on 3 A-levels. This might be A*AA; this means you must achieve that at the end of Year 13. Sometimes they do ask for 4, and 5 is extremely rare. Some people who have done A-levels already by the end of Year 12 might just get a single grade offer of "A*" in the remaining A-level they have to do. If you're doing Maths at a top university, they require that you take STEP. These are three 3-hour-long papers (STEP I, II, III) each marked out of 120. They are graded on a scale of S, 1, 2, 3, and U, where S is the highest grade. Typically a top university (e.g. Cambridge) will give you an offer of A*AA and 1,1 in STEP. The worst offer I have seen is somebody have A*A*A* + S,1 in STEP, but that was a rare occurrence. STEP is not to be underestimated; I have seen students get all A* at GCSE and A-level but end up with a 3,3 in STEP. It's easily one of the hardest maths papers you'll ever sit in your life if you've never done any olympiad stuff.

 

[neutrino']

thx fedex. I now have the info I need on british high school system. Thx

 

[jetwaterluffy]

The thing is, different countries have school systems, so sometimes it doesn't work out the same for the same age group. I think in Scotland they leave school with something similar to AS levels, while in america, they leave with the equivalent of GCSEs, I think. (I'm not sure, though. Might be something different.)

 

[hadsed]

marcalexander. if you know the famous particle physicist Richard Feynman, he had alredy mastered calculus by the age of 15 which means that it is time for you to start enlighten yourself of calculus. He discuses in his autobiography that his mastery of calculus at young age helped him at undergrad school (at MIT), which is clear that his success in QED and other researchs arises from his mastery of calculus. He studied calculus using the book "Calculus for the practical man" buy it or google it for ... as FeDeX_ LaTeX said a quick googling ;).

your consideration for your future career is appreaciable.

It's not a good idea to compare yourself to the greats. You will lose your confidence, you'll get frustrated at yourself for not having an easier time understanding, and also these books and stories about the greats don't tell you everything. No one ever talks about how hard Feynman worked on physics (if you can call it work, he very obviously loved to do it) and how much he thought about it. Not only that, but he was just a really, really smart guy. If you aren't as smart, that is okay, but trying to emulate his every move and then expecting that you'll do really really well is silly. Everyone is different, just try to learn as much as you can when you want to, making sure to have fun and enjoy yourself while you're doing it.

Marc, don't worry too much about it all. If you work hard in your math classes and try to learn, you'll be successful. In the meantime, you can try picking up some popular science books. This is what I did with quantum mechanics, and it's what really sparked my interest. Despite what everyone here says, reading these popular science books can help you understand a lot about theoretical physics, so when it comes time to learn the mathematics behind it, you can use the intuition and knowledge you've built to make sense of the math, and vice-versa. Also, you should subscribe/visit a site like Popular Science or Wired and check out the physics sections. These will keep you up to date about what's going on (kind of like reading papers, but much less difficult) as well as giving you a constant stream of stuff to look up and learn about. It'll keep your curiosity and interest going.

Most importantly, have fun.

 

[MarcAlexander]

It's not a good idea to compare yourself to the greats. You will lose your confidence, you'll get frustrated at yourself for not having an easier time understanding, and also these books and stories about the greats don't tell you everything. No one ever talks about how hard Feynman worked on physics (if you can call it work, he very obviously loved to do it) and how much he thought about it. Not only that, but he was just a really, really smart guy. If you aren't as smart, that is okay, but trying to emulate his every move and then expecting that you'll do really really well is silly. Everyone is different, just try to learn as much as you can when you want to, making sure to have fun and enjoy yourself while you're doing it.

Marc, don't worry too much about it all. If you work hard in your math classes and try to learn, you'll be successful. In the meantime, you can try picking up some popular science books. This is what I did with quantum mechanics, and it's what really sparked my interest. Despite what everyone here says, reading these popular science books can help you understand a lot about theoretical physics, so when it comes time to learn the mathematics behind it, you can use the intuition and knowledge you've built to make sense of the math, and vice-versa. Also, you should subscribe/visit a site like Popular Science or Wired and check out the physics sections. These will keep you up to date about what's going on (kind of like reading papers, but much less difficult) as well as giving you a constant stream of stuff to look up and learn about. It'll keep your curiosity and interest going.

Most importantly, have fun.

Well I'm certainly on the smart side of life for most 14 year olds. ;)

Thanks. I think I'll subscribe to the New Scientist magazine.

EDIT: I've actually started learning Calculus and I've got the hang of functions. After all, I'm an x-computer programmer(3 years of exp.).

 

[hadsed]

That's interesting (about programming). You should really continue with that. I started 'round about when you did, and I'd been thinking of doing CS my entire life onward till my last year in high school when I got to know some real physics. Still, programming has been a huge skill for me to get into doing research. I was able to start doing stuff right in the beginning of my first semester because I went to a guy who did computational astrophysics, which requires a ton of programming to run simulations. Now I'm working with two groups and publishing papers (in my second year now).

You should really, really keep with programming. I'd been on and off, but thankfully I'd done enough to keep my skills good enough and I kept learning. If I had stayed with it consistently, I'd probably be a programming grand master by now. So if you can be a theoretical physicist with incredible mathematical expertise as well as a programming guru... well, then you have a very bright future ahead of you. Not only that, but it gives you lots of flexibility with your future. You won't know in 10 years if you still want to do physics, and you certainly won't know your discipline or field that you'll be working in. Hell, you won't know if you even want to try and keep doing physics research after that.

 

[MarcAlexander]

That's interesting (about programming). You should really continue with that. I started 'round about when you did, and I'd been thinking of doing CS my entire life onward till my last year in high school when I got to know some real physics. Still, programming has been a huge skill for me to get into doing research. I was able to start doing stuff right in the beginning of my first semester because I went to a guy who did computational astrophysics, which requires a ton of programming to run simulations. Now I'm working with two groups and publishing papers (in my second year now).

You should really, really keep with programming. I'd been on and off, but thankfully I'd done enough to keep my skills good enough and I kept learning. If I had stayed with it consistently, I'd probably be a programming grand master by now. So if you can be a theoretical physicist with incredible mathematical expertise as well as a programming guru... well, then you have a very bright future ahead of you. Not only that, but it gives you lots of flexibility with your future. You won't know in 10 years if you still want to do physics, and you certainly won't know your discipline or field that you'll be working in. Hell, you won't know if you even want to try and keep doing physics research after that.

So you what you're saying is I should go back to developing my game engine?

 

[hadsed]

Not go back as in, quit physics and mathematics and just do that. I'm saying you should keep your skills sharp and try to learn new things as you go along. It might be just as beneficial to you as learning mathematics and physics. It doesn't matter what it is really, because it will be beneficial to you as long as you're challenging yourself. Of course you couldn't really do anything with programming and physics since most of the computational stuff that goes on in physics involves solving math problems that are too hard or take too long to solve on paper.

So, short answer, yes you should.