Studying How to self-study physics past High School Level?

AI Thread Summary
A teenage physics enthusiast seeks guidance on advancing their knowledge beyond A-level physics. They have studied various textbooks and completed numerous practice problems but are unsure how to proceed. The discussion emphasizes the importance of parallel development in mathematics and physics, particularly the necessity of mastering calculus, including vector calculus, before delving deeper into physics topics like electromagnetism and astronomy. Recommendations include following established university curricula, such as MIT's or Yale's free online courses, and utilizing specific textbooks like Halliday, Resnick, and Krane for foundational physics. Participants suggest exploring hands-on projects related to electromagnetism and astronomy to enhance practical understanding. Additionally, resources for learning mathematics relevant to physics are highlighted, with an emphasis on avoiding overly abstract mathematical texts. Overall, the conversation encourages a structured approach to learning, integrating both theoretical and practical aspects of physics.
  • #51
TensorCalculus said:
This is going on my "favourite quotes of all time" list 😆

What - really? I mean I guess they're doing similar things, but surely they're at least somewhat different? I see people operating on entire tensors at once (like I do when I do coding sometimes, except the computer's doing all the hard work for me in that case) - is that actually the same thing as integrating over a vector field (such as when looking at line integrals, surface integrals etc in things such as Gauss's law and Ampere's law?)

Sounds good! I might take a look even if it's aimed at mathematicians, Physics may be my passion (and I do therefore steer away from abstract maths) but I am a huge maths nerd too and do enjoy maths just for the fun of it :D
> What - really?

Yeah, it was a surprise to me too! If you're familiar with tensors then you should know that they're classified as covariant and contravariant tensors as well as into symmetric and antisymmetric tensors. We also differentiate between tensors and tensor fields in the same way that vectors are different from vector fields. Often tensor fields are just called tensors. Differential forms are precisely covariant and antisymmetric tensor fields.

The study of differential forms and manifolds differs from that of tensors in physics in that they use coordinate free formalisms and hence they come without indices though you can introduce indices if you want to work locally.

>is that actually the same thing as integrating over a vector field (such as when looking at line integrals, surface integrals etc in things such as Gauss's law and Ampere's law?)

In differential geometry it is natural to integrate differential forms over a manifold. This is because we can define it using only the smooth structure and without using a metric. It's one of the important uses of differential forms. This subsumes the notion of line, surface and volume integrals into one formalism. The reason why we can integrate vector fields over curves, surfaces and volumes in vector analysis is to do with the implicit use of the metric here. Also Greens's theorem, the Kelvin-Stokes theorem and the divergence theorems are all aapects of one theorem in differential geometry which is called Stokes theorem.

>Sounds good! I might take a look even if it's aimed at mathematicians ...

In that case I'd recommend the volume on smooth manifolds as that covers differential forms and also the volume on Riemannian geometry as that covers the geometry used in general relativity. I'd also dip into volume one for the definition of a topological manifold as that defines a manifold as simply as possible - it's just a space that looks locally like a Euclidean space. I've also said this before but I'll repeat it because its such an excellant book, have a look into Baez & Muniain - Gauge Fields, Knots & Gravity as they teach differential geomeyry whilst keeping an eye on the physics its used for.
 
  • Informative
Likes TensorCalculus
Physics news on Phys.org
  • #52
Albertus Magnus said:
I haven't checked on these specific titles, but I often buy physics texts at thrift books or on Ebay, for example I just got an older edition but still perfectly useful copy of Gradshteyn and Rhyzik $20.00 on ebay, whereas a new copy would have cost me about $200.00.
I've checked for used copies too - my go-to is Ebay. Nothing below £70 :(
Albertus Magnus said:
A copy of Haliday and Resnick from the 1970s would be just about as good as a more updated copy, and I bet it would be quite affordable. Foundational type subjects like classical mechanics don't change too fast, however, subjects like astronomy on the other hand have developed so much in the last 20 years that it pays to buy newer texts.
True - I'll take a look then. Hopefully they won't be above £20!
Albertus Magnus said:
I just wanted to add: the Feynmann Lectures on Physics are available free online and they are awesome! He covers classical mechanics, electromagnetism and quantum While teaching some maths along the way. This would be a good place to get some of that vector calculus.
I've already read them (well... volumes 1 and 2). They are brilliant - you're right!
jtbell said:
For self study, there's no need to buy the current edition. Used copies a few editions back, whatever you can find cheaply, are fine for that purpose.

In the US at least, instructors usually assign homework exercises out of the textbook, and often assign specific pages or sections for reading. Publishers change up the book's layout, exercises, etc. in new editions, in order to discourage students from buying used books from others who have already taken the course. At the introductory level, many students are not physics majors, will not be studying physics further, see no reason to keep the book as a reference, and therefore happy to sell their copies if they can.
That's what I do - but I don;t think as someone from the UK I'll be able to get my hand on any American physics majors (I don't think textbooks are used the same way here... maybe. If they are, I have a few people I know studying physics in Uni, I might be able to borrow some books form them)
Muu9 said:
Halliday Resnick Walker is roughly at the same level as University Physics by Young and Freedman, so I wouldn't bother with the former for someone who has completed the latter.

Here's another idea: ask university libraries in driving distance when they will update their copies of physics/math texts, which usually involves throwing out the old editions, so you can pick them up for dirt cheap or free.
Oh - I thought Haliday Resnick Walker might be Harder. I did love Young and Freedman though, and learnt a lot, so if I can get a copy of Haliday Resnick Walker at a reasonable price, I'll read it - I think it might have some valuable stuff for me (hopefully). I get where you're coming from though - so I probably won't study it extensively or anything. About the library thing - that seems like a good idea. I'll find their contact :D
Mozibur Rahman Ullah said:
Yeah, it was a surprise to me too! If you're familiar with tensors then you should know that they're classified as covariant and contravariant tensors as well as into symmetric and antisymmetric tensors. We also differentiate between tensors and tensor fields in the same way that vectors are different from vector fields. Often tensor fields are just called tensors. Differential forms are precisely covariant and antisymmetric tensor fields.
I mean... I know of this -> (If you're familiar with tensors then you should know that they're classified as covariant and contravariant tensors as well as into symmetric and antisymmetric tensors.) but I don't know anything about doing maths with tensor fields...
Mozibur Rahman Ullah said:
In differential geometry it is natural to integrate differential forms over a manifold. This is because we can define it using only the smooth structure and without using a metric. It's one of the important uses of differential forms. This subsumes the notion of line, surface and volume integrals into one formalism. The reason why we can integrate vector fields over curves, surfaces and volumes in vector analysis is to do with the implicit use of the metric here. Also Greens's theorem, the Kelvin-Stokes theorem and the divergence theorems are all apects of one theorem in differential geometry which is called Stokes theorem.
Ahhhh - that makes so much sense, thanks! I never really thought about what I was doing when integrating vector fields over things such as curves - thank you for the insight!
Mozibur Rahman Ullah said:
In that case I'd recommend the volume on smooth manifolds as that covers differential forms and also the volume on Riemannian geometry as that covers the geometry used in general relativity. I'd also dip into volume one for the definition of a topological manifold as that defines a manifold as simply as possible - it's just a space that looks locally like a Euclidean space. I've also said this before but I'll repeat it because its such an excellant book, have a look into Baez & Muniain - Gauge Fields, Knots & Gravity as they teach differential geomeyry whilst keeping an eye on the physics its used for.
Alright! I'll take a look. I might work on my vector calc first though since it's not particularly strong. Thank you for the recommendation!
 
  • #53
TensorCalculus said:
Hello! So I'm a teenage physics enthusiast, who wants to take my knowledge past A-level (or in America I believe this would be high school level) physics.
I've studied multiple textbooks like Young and Freedman's University physics, studied maths from books like mathematical methods for physics and engineering.
I solidified all that by doing lots (like, LOTS) of the practice problems and some Olympiad papers.

I don't really know where to go from here.
I've resorted to surfing the internet, and finding free courses and watching YouTube videos that satisfy my interest, or reading popular science, reading high-school textbooks, and just sitting around.
I'm not entirely sure what books to buy - I fear that if I accidentally skip straight to something too advanced I'll not have strong basics. My teacher told me to just read popular science, but I really enjoy looking at the math behind things and I feel that there are few popular science books (that I've read at least) that satisfy my curiosity.
I do really, really love physics, and want to try and study it further - unfortunately, I'm 13 and have quite a long time till I can study physics in University/College.
Does anyone have any resources they would recommend to me (my interest particularly lies around astronomy and electromagnetism) , or any advice to give?
You are already doing it by visiting this website, :) there are tutorials, papers, and articles about almost every branch of physics.
But if you want something more "formal" and can be measured for 3rd parties (jobs and universities), I'd do a course on coursera, edX, or any reputable online teaching website.
 
  • Like
Likes TensorCalculus and Albertus Magnus
  • #54
sairoof said:
You are already doing it by visiting this website, :) there are tutorials, papers, and articles about almost every branch of physics.
Yep! And since I've joined, I've really enjoyed looking at all of it too :D
sairoof said:
But if you want something more "formal" and can be measured for 3rd parties (jobs and universities), I'd do a course on coursera, edX, or any reputable online teaching website.
Oh - taking courses online can help for jobs and Universities? I never even considered the possibility!
 
  • #55
TensorCalculus said:
Yep! And since I've joined, I've really enjoyed looking at all of it too :D

Oh - taking courses online can help for jobs and Universities? I never even considered the possibility!
To a certain degree. Not every company or country take these courses seriously, so see what's the situation in your area and then decide if they are worth the investment or not.
But also, a lot of these courses are actually free and you only need to pay if you want the certificate.
 
  • Informative
Likes TensorCalculus
  • #56
sairoof said:
To a certain degree. Not every company or country take these courses seriously, so see what's the situation in your area and then decide if they are worth the investment or not.
But also, a lot of these courses are actually free and you only need to pay if you want the certificate.
Hmm... wow. I've taken my fair share of course on MIT Open Courseware (and recently, have been doing some form Yale etc as they were recommended on this feed). I wonder if they could be put on various applications. I'll check if they care about it much here in Britain.
Thank you for suggesting this!
 
  • #57
The internet can be a dangerous place to learn something. You easily get distracted. Videos - in my opinion - sell the illusion of understanding without reassuring that insights were provided. You watch them, agree, and forget about them quicker than it took to watch them. I don't think that such a passive way of study can ultimately lead to insights you would alternatively gain by working through a subject with a lot of paper and ink. You already lost a lot of time by commenting here. If you were really interested, you would use every spare minute to learn more about a subject. The way from school to relativity theory and quantum mechanics is a long, winding one. The necessary mathematics alone is complicated and all but trivial. Curiosity should be your major stimulus. And then, it is relatively irrelevant what your resources are as long as they are from a university, a lecture note, or a good textbook.
 
  • Informative
Likes TensorCalculus
  • #58
fresh_42 said:
The internet can be a dangerous place to learn something. You easily get distracted. Videos - in my opinion - sell the illusion of understanding without reassuring that insights were provided. You watch them, agree, and forget about them quicker than it took to watch them. I don't think that such a passive way of study can ultimately lead to insights you would alternatively gain by working through a subject with a lot of paper and ink.
Well... fair enough.
fresh_42 said:
You already lost a lot of time by commenting here. If you were really interested, you would use every spare minute to learn more about a subject.
Yeah... I mean I tend to comment on little 5-10 minute pockets of time in school where I can't really do much else...
fresh_42 said:
The way from school to relativity theory and quantum mechanics is a long, winding one. The necessary mathematics alone is complicated and all but trivial. Curiosity should be your major stimulus. And then, it is relatively irrelevant what your resources are as long as they are from a university, a lecture note, or a good textbook.
Fair (again). I think I've got all the answers that I've wanted, and a wealth of resources and advice from all of the wonderful people on PF! My goal was never relativity or QM, but I do understand that the path is long. I'll take some time to work through all the resources everyone has kindly provided me with, (and keep in mind all the advice!) Thank you everyone :smile:
 
  • #59
fresh_42 said:
The internet can be a dangerous place to learn something. You easily get distracted. Videos - in my opinion - sell the illusion of understanding without reassuring that insights were provided. You watch them, agree, and forget about them quicker than it took to watch them. I don't think that such a passive way of study can ultimately lead to insights you would alternatively gain by working through a subject with a lot of paper and ink. You already lost a lot of time by commenting here. If you were really interested, you would use every spare minute to learn more about a subject. The way from school to relativity theory and quantum mechanics is a long, winding one. The necessary mathematics alone is complicated and all but trivial. Curiosity should be your major stimulus. And then, it is relatively irrelevant what your resources are as long as they are from a university, a lecture note, or a good textbook.
I agree that the internet can be dangerous but I think it can also be valuable. I learnt a lot from Cohl Furey's set of short videos on the octonions and their use in studying the structure of the standard model, I also learnt a lot about spinors from Eigenchris's set of video lectures on spinors. Another series of short videos which I found fascinating were the Catster's series of 10 minute videos on various aspects of category theory. I found all of these video lectures on youtube. On the other hand there are plenty of books I wished I hadn't wasted time on. The moral of this story that one has to be discerning. And its easier when the discerning has already been done by a curator. At university this is done by a recommended reading list. Certainly were I teaching a university course those lectures I alluded to above would be on the recommended viewing list.
 
  • Like
Likes WWGD, TensorCalculus and sairoof
  • #60
Mozibur Rahman Ullah said:
I learnt a lot from Cohl Furey's set of short videos on the octonions and their use in studying the structure of the standard model, I also learnt a lot about spinors from Eigenchris's set of video lectures on spinors. Another series of short videos which I found fascinating ...
This is likely a matter of personal taste and might not be generally applicable. It might also depend on whether videos are used exclusively or additionally. My major concern is the passivity they bring with them. I think science cannot be learned by consumption, and that it has to be actively worked out. You have to make mistakes and follow wrong paths in order to see possible traps, and there are no mistakes and wrong paths in videos. However, I admit that this is a personal opinion based on the fact that I have to write down something if I want to memorize it reliably, and you may have a different experience and opinion.
 
  • Like
Likes TensorCalculus
  • #61
fresh_42 said:
The internet can be a dangerous place to learn something. You easily get distracted. Videos - in my opinion - sell the illusion of understanding without reassuring that insights were provided. You watch them, agree, and forget about them quicker than it took to watch them. I don't think that such a passive way of study can ultimately lead to insights you would alternatively gain by working through a subject with a lot of paper and ink. You already lost a lot of time by commenting here. If you were really interested, you would use every spare minute to learn more about a subject. The way from school to relativity theory and quantum mechanics is a long, winding one. The necessary mathematics alone is complicated and all but trivial. Curiosity should be your major stimulus. And then, it is relatively irrelevant what your resources are as long as they are from a university, a lecture note, or a good textbook.
There's a risk associated with almost everything.
But as mozibur mentioned there's a lot of value to be found on the internet. A lot of skills can be learned from YouTube lectures and I personally learned how to assemble and fix watches with simple tools using the internet.
and I agree, mindless consumption of science is not the way to do it.
 
  • Like
Likes TensorCalculus and WWGD
  • #62
sairoof said:
There's a risk associated with almost everything.
But as mozibur mentioned there's a lot of value to be found on the internet. A lot of skills can be learned from YouTube lectures and I personally learned how to assemble and fix watches with simple tools using the internet.
and I agree, mindless consumption of science is not the way to do it.
Mindless consumption of anything other than possibly recreation and entertainment is not conducive to good results.
 
  • Like
Likes TensorCalculus and sairoof
  • #63
fresh_42 said:
This is likely a matter of personal taste and might not be generally applicable. It might also depend on whether videos are used exclusively or additionally. My major concern is the passivity they bring with them. I think science cannot be learned by consumption, and that it has to be actively worked out. You have to make mistakes and follow wrong paths in order to see possible traps, and there are no mistakes and wrong paths in videos. However, I admit that this is a personal opinion based on the fact that I have to write down something if I want to memorize it reliably, and you may have a different experience and opinion.
True, but you can complement with ChatGpt, while fact-checking what it says. The back-and-forths with it can be intense.
 
  • #64
fresh_42 said:
My major concern is the passivity they bring with them. I think science cannot be learned by consumption, and that it has to be actively worked out. You have to make mistakes and follow wrong paths in order to see possible traps, and there are no mistakes and wrong paths in videos.
That is 150% true, and why I agree with your point about the videos. Of course, I'm not experienced in learning things, but so far out of all the ways of learning I've tried and tested, I've learnt that just passively sitting there and consuming content without following it up with some action of my own (even if that's just writing down the key points, or following it up with research further), or doing things like reading notes, have never really worked - no matter how interesting I find the content.
sairoof said:
But as mozibur mentioned there's a lot of value to be found on the internet. A lot of skills can be learned from YouTube lectures and I personally learned how to assemble and fix watches with simple tools using the internet.
and I agree, mindless consumption of science is not the way to do it.
I am a fan of watching maths/physics Youtube videos on my way to school and in history lessons. I also watch lectures then follow up by doing questions that apply what I've learnt. For me that's not mindless consumption - and you're right that it brings value
WWGD said:
True, but you can complement with ChatGpt, while fact-checking what it says. The back-and-forths with it can be intense.
I'm genuinely terrified of using ChatGPT. I used to use copilot to check my answers to problems when the textbook didn't provide them, and the utterly obscene amount of times it got even the most basic physics wrong put me off from using AI for maths and physics pretty much permanently. I use AI to find sources but that's pretty much it. How do you use it?
 
  • #65
TensorCalculus said:
That is 150% true, and why I agree with your point about the videos. Of course, I'm not experienced in learning things, but so far out of all the ways of learning I've tried and tested, I've learnt that just passively sitting there and consuming content without following it up with some action of my own (even if that's just writing down the key points, or following it up with research further), or doing things like reading notes, have never really worked - no matter how interesting I find the content.

I am a fan of watching maths/physics Youtube videos on my way to school and in history lessons. I also watch lectures then follow up by doing questions that apply what I've learnt. For me that's not mindless consumption - and you're right that it brings value

I'm genuinely terrified of using ChatGPT. I used to use copilot to check my answers to problems when the textbook didn't provide them, and the utterly obscene amount of times it got even the most basic physics wrong put me off from using AI for maths and physics pretty much permanently. I use AI to find sources but that's pretty much it. How do you use it?
As a funnel, asking more and more specific questions. I've used Copilot and, while I received some answers that were wrong, many were right. I test drove it asking it questions I knew the answer to. Edit:
https://mathforums.com/t/does-bing-...-it-possible-to-make-it-more-reliable.370376/
 
  • Informative
Likes TensorCalculus
  • #67
WWGD said:
As a funnel, asking more and more specific questions. I've used Copilot and, while I received some answers that were wrong, many were right. I test drove it asking it questions I knew the answer to. Edit:
https://mathforums.com/t/does-bing-...-it-possible-to-make-it-more-reliable.370376/
Ah. I see. Not as an answer checker. I found that copilot got the answers wrong at least 40% of the time, and that I'd have to then check/cross-reference the validity myself, at which point I might as well just mark my answers via checking multiple times/cross referencing methods with similar problems. I might try using chatgpt for general questions, if I'm stuck on any of the courses/readings I've picked up recently
WWGD said:
Sorry, Ill elaborate on this later.
don't apologise! :D
 
  • #68
I hear Claude 3.7 sonnet is currently the strongest model; make sure to use the extended thinking option
 
  • #69
I haven't tried it - it's blocked on our school laptops :(
does it work on phones?
 
  • #70
WWGD said:
As a funnel, asking more and more specific questions. I've used Copilot and, while I received some answers that were wrong, many were right. I test drove it asking it questions I knew the answer to.
That is a problem. If you do not know how to tell what is what then it is ultimately useless! De-learning is a terrible job! Even Wikipedia is a better reference than any AI.
 
  • Like
Likes TensorCalculus
  • #71
TensorCalculus said:
I haven't tried it - it's blocked on our school laptops :(
does it work on phones?
Yes, there is an app and a website
 
  • Like
Likes TensorCalculus
  • #72
fresh_42 said:
That is a problem. If you do not know how to tell what is what then it is ultimately useless! De-learning is a terrible job! Even Wikipedia is a better reference than any AI.
I was able to tell what's what. In my experience it is statistically better than other methods, more portable, easier to implement. The process of " funneling" itself is helpful and just straight critical thinking. But to each their own choice of learning methods, strategies.
 
  • #73
fresh_42 said:
That is a problem. If you do not know how to tell what is what then it is ultimately useless!
Yeah - I have to agree on this. I've used AI to help me with subjects other than maths and physics, but as soon as it comes to me asking about physics, it gets it really really wrong half the time, and I have to fact check it myself by looking at other sources on the internet which say the same answer, at which point I might as well have just not used the AI and researched it myself.
fresh_42 said:
De-learning is a terrible job! Even Wikipedia is a better reference than any AI.
What's wrong with Wikipedia? I know it's edited by lots of people, but I find it quite useful.
Muu9 said:
Yes, there is an app and a website
Ok, I've just tried it and it does seem to be more intelligent than Copilot at least (which is a good sign)
WWGD said:
I was able to tell what's what. In my experience it is statistically better than other methods, more portable, easier to implement. The process of " funneling" itself is helpful and just straight critical thinking. But to each their own choice of learning methods, strategies.
I'm not able to tell what's what - that's the problem. The only wrong answers I can spot are orders of magnitude off. Do you have an example of funneling?

Also - I managed to get my hands on a copy of Haliday resnick walker! (though it's only arriving in about a week's time)
I've also spent as much free time and sleep as possible going through the MIT calculus courses (and the yale physics ones - though less so on them) - I've gotten a fair way through! They've been really really helpful and I've learnt a lot! 😃
 
  • #74
TensorCalculus said:
Yeah - I have to agree on this. I've used AI to help me with subjects other than maths and physics, but as soon as it comes to me asking about physics, it gets it really really wrong half the time, and I have to fact check it myself by looking at other sources on the internet which say the same answer, at which point I might as well have just not used the AI and researched it myself.

What's wrong with Wikipedia? I know it's edited by lots of people, but I find it quite useful.

Ok, I've just tried it and it does seem to be more intelligent than Copilot at least (which is a good sign)

I'm not able to tell what's what - that's the problem. The only wrong answers I can spot are orders of magnitude off. Do you have an example of funneling?

Also - I managed to get my hands on a copy of Haliday resnick walker! (though it's only arriving in about a week's time)
I've also spent as much free time and sleep as possible going through the MIT calculus courses (and the yale physics ones - though less so on them) - I've gotten a fair way through! They've been really really helpful and I've learnt a lot! 😃
I'm referring to using all of these combined, and complementary to each other.
 
  • Like
Likes TensorCalculus
  • #75
TensorCalculus said:
What's wrong with Wikipedia? I know it's edited by lots of people, but I find it quite useful.
There is nothing wrong with it, as long you won't be confronted with the admins there - then it becomes politics with all its ugly faces. Wikipedia is not always as accurate as it should be, especially in non-science subjects but I also found errors on mathematical pages. I like it because of its references and the possibility of switching languages. The different language versions are not simply translations, they have different content. You don't even need to speak a language fluently to read about subjects with many formulas or words of Latin origin. This reduces possible biases. It also depends on what you are looking for. Wikipedia cannot replace a textbook or lecture note, as it usually does not contain derivations. It's more a collection of quick references than a collection of treatises.
 
  • Like
Likes WWGD and TensorCalculus
  • #76
Pages can be modified by anyone with an Internet connection. Edit: The material in the page may "Converge" to the correct answer, but it's not clear if/when.
 
  • #77
@Muu9 - are you paying for claude? It's not letting me use the extended thinking mode otherwise... Edit: Even without extended thinking mode, it's doing really really well. No incorrect answers so far and the problem difficulty is decent. Not to mention the debugging skills when coding - I love the fact that I can link it to my github :D
 
  • #78
Deepseek allows for deep thinking in the free tier
 
  • Like
Likes TensorCalculus
  • #80
  • #81
To highlight the difference between differential geometry and vector analysis, it's worth noting that in vector analysis we have the four equations of Maxwell's equations (in Maxwell's original description there were in the order of twenty equations). In differential geometry there are two:



$$dF = 0$$

$$*d*F = J$$



Here ##F## is the electromagnetic strength, ##J## is the current whilst ##d## is the exterior derivative which generalises ##grad##, ##curl## & ##div## and ##*## is the Hodge star. I hope you agree not only have we reduced the equations in number but they also look simpler. But more is true - Maxwell's equations are true only for 3d Euclidean space whereas these equations are true for any curved space and for any dimension.

Now Yang-Mills theory is a generalisation of Electromagnetism and it models the weak and strong force. In differential geometry the two equations above generalise to the Yangs-Mill's equations:



$$d^DF = 0$$

$$*d^D*F = 0$$



Here ##d^D## is the exterior covariant derivative which further generalises the exterior derivative ##d## by combining it with a covariant derivative ##D##. Notice just how similar it looks like the equations for Electromagnetism. I hope this helps clarify why differential geometry is worth pursuing for its use in physics.
 
Last edited by a moderator:
  • Like
Likes dextercioby and TensorCalculus
  • #82
Yes - I see what you mean!
Mozibur Rahman Ullah said:
I hope you agree not only have we reduced the equations in number but they also look simpler. But more is true - Maxwell's equations are true only for 3d Euclidean space whereas these equations are true for any curved space and for any dimension.
This is actually really, really cool (you have convinced me to learn differential geometry just by showing me this haha). Thank you for showing it to me :D
Mozibur Rahman Ullah said:
Here dD is the exterior covariant derivative which further generalises the exterior derivative d by combining it with a covariant derivative D.
I don't really understand what this means though - do you mind explaining it to me? I've tried to do some research to clarify on the internet, but I just ended up getting bogged down in endless articles that I don't understand, AI explanations that don't really answer the question and more and more difficult maths :(
 
  • Like
Likes Mozibur Rahman Ullah
  • #83
@TensorCalculus . I posted early in this thread. I see that it's grown to >80 posts. From a quick scan, many posts concern advanced topics. But at this stage, you should be concentrating on core fundamentals. One step at a time. That's why I recommended early on that you follow an established university syllabus, rather than being directed in a zillion directions. FWIW I got my PhD in physics many moons ago and never touched differential geometry. But I preferred spending my time having fun in the lab.
 
  • Like
Likes TensorCalculus and gleem
  • #84
CrysPhys said:
@TensorCalculus . I posted early in this thread. I see that it's grown to >80 posts. From a quick scan, many posts concern advanced topics.
Really? I thought it was just the differential geometry? What else is there?
CrysPhys said:
But at this stage, you should be concentrating on core fundamentals. One step at a time. That's why I recommended early on that you follow an established university syllabus, rather than being directed in a zillion directions.
You're right - I completely agree - and that's why I listened to you and @Muu9 - I've been doing the courses from MIT and Yale. I'm a very decent way through the first one I decided to do which was on single variable calculus. As for all the other suggestions, I've taken it all on board and made a list of things I want to do, in order of increasing difficulty
CrysPhys said:
FWIW I got my PhD in physics many moons ago and never touched differential geometry. But I preferred spending my time having fun in the lab.
Really?!!!! Oh. But it looked so cool - and I've seen it being mentioned before in various physics... things before. I'd want to go into astrophysics later on (at least, right now I would want to, who knows I might change my mind) so differential geometry would probably be useful (relativity and all that)... right???
 
  • #85
TensorCalculus said:
Really?!!!! Oh. But it looked so cool - and I've seen it being mentioned before in various physics... things before. I'd want to go into astrophysics later on (at least, right now I would want to, who knows I might change my mind) so differential geometry would probably be useful (relativity and all that)... right???
I'm not qualified to answer that. I specialized in experimental solid-state physics and never needed it for my particular research. Others can chime in where it would be needed or helpful. But again, you've got a waay lot more fundamental core physics to learn; so don't get distracted and side-tracked.
 
  • Like
Likes TensorCalculus
  • #86
CrysPhys said:
But again, you've got a waay lot more fundamental core physics to learn; so don't get distracted and side-tracked.
genuinely - thank you so much for this, I need to hear this :D
I always get carried away learning things too early and then messing up all of my basics, all the time, and I'm terrified of it:
TensorCalculus said:
I fear that if I accidentally skip straight to something too advanced I'll not have strong basics.
^ as in my original query
(though I have noted down @Mozibur Rahman Ullah 's suggestions - particularly the books that were strongly recommended. @CrysPhys may have reminded me of something that I desperately needed reminding of, but I did a bit of digging after the recommendation and differential geometry looks like something I would want to study - even if it weren't useful in physics (which, it quite clearly, can be), the maths seems awesome. I do think that I have a bit of time till I'm ready for it though, I think there is a point there... )
 
  • #87
TensorCalculus said:
Yes - I see what you mean!

This is actually really, really cool (you have convinced me to learn differential geometry just by showing me this haha). Thank you for showing it to me :D

I don't really understand what this means though - do you mind explaining it to me? I've tried to do some research to clarify on the internet, but I just ended up getting bogged down in endless articles that I don't understand, AI explanations that don't really answer the question and more and more difficult maths :(

Great, I'm glad you feel its worth learning. But I'd hold your horses about learning this stuff seriously at this stage of your education. This is advanced stuff and the exterior covariant derivative is really advanced stuff! Typically, you will learn vector analysis in undergraduate courses and differential geometry in graduate courses and the exterior covariant derivative is most likely taught at a second course at this level. Certainly I wasn't introduced to it in my masters course on theoretical physics. Nor does the textbook I recommended, Lee's Smooth Manifolds explains this. I first learnt about it in Baez & Muniain's Gauge Fields, Knots and Gravity which I have already recommended. It takes the physics approach to differential geometry and so avoids all the very careful constructions in math whilst still being relatively rigorous. The chapter you should look at is chapter II - 3, Curvature and the Yang-Mills equation, pg.250. But I urge you to study the text systematically, or at least go through it trying to understand the concepts that they introduce to get a sense of the terrain. It pays to go through the text step by step. But I also think at this stage of your education you should be getting the fundamentals of physics down - that is really, really important - and not be distracted by advanced and really advanced stuff. I promise you they will still be waiting for you later on in your career. It's a problem that I've had in my own education, so I feel your frustrations. I explained what I did in my previous post as motivation that this stuff is worth learning and not as an indication that you should be learning this stuff at this stage of your education.

Nevertheless, I've written an exposition below which I hope will give you a sense of the terrain without getting bogged down in detail. I hope it helps without further confusing you and leading you astray!

First, to really get grips with the exterior covariant derivative you need to be comfortable with manifolds, tangent bundles and vector bundles as well as their spaces of sections - basically tangent fields and vector fields, as well as the wedge product (a generalisation of the cross product which works for all dimensions, the cross product only works in 3d) and tensor product and of course differential forms and the exterior derivative, oh as well as understanding that tangent fields can also be interpreted as certain differential operators. This is a long list of mathematical technology, so I've written an exposition that gives you a birds-eye view of some of the neccessary concepts without getting into the detail that's needed to really grasp these ideas and make use of them.


The first thing to learn is manifolds. This is basically a formalisation of one aspect of the notion of covariance. In fact Einstein said:


"So there is nothing for it but to regard all imaginable systems of coordinates, on principle, as equally suitable for description of nature"


Manifolds are spaces that are equipped with charts, that is coordinate systems. They are also equipped with transition functions that map between these different charts. This sounds complicated, but it's not so bad as we learn to manipulate manifolds. For example, if you had two manifolds ##M## & ##N## then you can multiply them ##M \times N## and you can take their disjoint sum ##M \sqcup N##. For example if we took the real line is a manifold as well as the circle. If we multiply them we get the infinite cylinder and if we add them we just get the line 'next' to the circle. So we have a calculus of manifolds. It's also important to distinguish between manifolds by themselves and manifolds that are embedded in Euclidean space. For example, if you imagine a sphere - by the way, mathematicians call the surface of a ball, a sphere and the interior of a ball, is just called a ball, then you will probably imagine a sphere in space. This makes it easy to see what the tangent planes on a sphere are. This is a sphere embedded in space. When a mathematician refers to a sphere, it is a sphere all by itself, sort of hanging in the void. This makes it difficult to understand what the tangent planes are as there's nowhere we can take the tangent plane. There's only the void surrounding the sphere. However, there is a way of introducing the tangent planes. This relies on maths but a quick way of seeing this, is simply to invoke a theorem that all manifolds are embeddable in a Euclidean space of high enough dimension and then just take the tangent planes as usual. The manifold with all the tangent spaces attached is called the tangent bundle over that manifold.


Now if ##M## is a manifold - for example a sphere - then ##TM## is the usual notation for the tangent bundle over ##M##. The map ##T## is called the tangent functor and this has nice properties. For example, if ##M## & ##N## are manifolds then ##T(M \times N) = TM \times TN## and obviously ##T(M \sqcup N) = TM \sqcup TN##. This is already important. For example, the tangent space to the infinite cylinder can now be easily calculated. It is ##T(S^1 \times \mathbb{R}) = T(S^1) \times T(\mathbb{R})##.


It also turns out that if we have maps ##f: M \rightarrow N## and another map ##g: N \rightarrow P##, then ##T(g\circ f) = Tg \circ Tf##. To understand this intuitively, it best to see the situation geometrically. When we map one manifold to another, then we also map the tangent planes on the first manifold to the second manifold. Then if we have a composition of maps then the tangent maps also compose. It's best to see this as an illustration but I don't know if Physics Forums supports images. Its also worth mentioning that this rule is basically the chain rule we learn to love in calculus, generalised to multivariable calculus and then to manifolds. The way to show this is to work locally, that is by fixing charts and then working out the various expressions.


Now the covariant derivative ##D## is a replacement for the directional derivative in differential geometry. The usual notion of a directional derivative doesn't work as it relies on the embedding of the manifold in Euclidean space. However, that theorem of embedding the manifold in a Euclidean space of high enough dimension comes in useful again. It turns out that the covariant derivative is the orthogonal projection of the directional derivative onto the tangent space. I say tangent space rather than tangent plane because whilst a 2d manifold has a 2d tangent space - aka a tangent plane - a n-dimensional manifold has a n-dimensional tangent space. Usually, when the covariant derivative is introduced it is done intrinsically but I find this geometric way of defining it more intuitive.


We need the covariant derivative because the exterior covariant derivative combines the exterior derivative and the covariant derivative. I haven't introduced the exterior derivative but we can at least see how the exterior covariant derivative is defined on Baez & Muniain, pg.250


We are going to go for an inductive definition. This is because we have differential 0-forms, 1-forms, 2-forms all the way upto ##n##-forms where ##n## is the dimesion of the manifolds. A differential k-form is said to have order ##k##. The space of k-forms on a manifold ##M## is denoted by ##\Omega^k(M)##. It will turn out that the exterior derivative of a k-form will result in a form of one degree higher, aka a k+1-form. In symbols:


##d: \Omega^k(M) \rightarrow \Omega^{k+1}(M)##


We really ought to have a superscript ##k## on the ##d## but this is just usually left implicit to avoid notational clutter.


Now, its important to realise 0-forms on a manifold are precisely real valued functions on the manifold. Then the exterior derivative ##d## of a differential 0- form - aka a function ##f## - is defined by:


##df(v) := v(f)##


Here ##v## is a vector field, really a tangent field. Now I've already said above that tangent fields can be reinterpreted as a certain differential operator. So ##v(f)## is saying ##v## is deriving ##f##.


Now the exterior covariant derivative is defined by introducing an auxilary covariant derivative ##D## and this must live on some vector bundle ##E##. We haven't introduced vector bundles before. They are a generalisation of tangent bundles. And just as tangent bundles have a covariant derivative, so do vector bundles. Now instead of having k-forms we have instead E-valued k-forms. The space of all E-valued k-forms is written as ##\Omega^k(M, E)##. Then the exterior covariant derivative maps E-valued k-forms to E-valued k+1-forms. Or in symbols:


##d^D : \Omega^k(M, E) \rightarrow \Omega^{k+1}(M, E)##


Here again the ##d^D## ought to have a superscript ##k##, but its left implicit. Now to run the inductive definition of ##d^D## we need to find out what E-valued 0-forms are. These are just vector fields of E. We denote them by latin letters, so in the following formula for ##d^D##, by the letter s.


##(d^Ds)(v) = D_vs##


Thus we see in the zeroth step of the inductive definition we see we have replaced the derivative ##v## acting on ##f## by ##D_v## on ##s##. Here ##f## is a 0-form, aka a function, and ##s## is an E-valued 0-form, aka a vector field in ##E##.


We then inductively define ##d## for higher differential forms via a variation of the Liebniz rule for the derivative of products (Baez & Muniain, pg.63) :


##d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta##


Here ##k## is the order of the differential form ##\alpha##. The symbol ##\wedge## is the wedge or exterior product - it's the generalisation of the cross product to any dimension because the usual cross product just works in 3d only. The corresponding inductive definition for the exterior covariant derivative is on Baez & Muniain, pg.250. It is also a variation on the Liebniz product rule:


##d^D(s \otimes \alpha) = d^Ds \wedge \alpha + s \otimes d\alpha##


Here ##s## is an E-valued 0-form and ##\alpha## is an E-valued k-form. Here the symbol ##\otimes## is the tensor product.


It's been a long post but I hope this gives you some sense on how the exterior covariant derivative comes about. To be honest, I don't expect you to grasp all this as I've skipped over a lot of detail - for example, what is the wedge product. And don't feel frustrated if it escapes you, as I've already said the exterior covariant derivative is really, really advanced stuff. But I do hope what I've written gives you a feeling for the shape of the terrain you would have to cover to rigourously learn this stuff. If you have any questions on what I've written then feel free to ask.
 
Last edited:
  • Informative
Likes TensorCalculus
  • #88
Mozibur Rahman Ullah said:
But I'd hold your horses about learning this stuff seriously at this stage of your career.
<<Emphasis added.>> The OP is 13! :rolleyes:
 
  • #89
CrysPhys said:
<<Emphasis added.>> The OP is 13! :rolleyes:
I meant to say at this stage of your education! Good catch.
 
  • #90
CrysPhys said:
<<Emphasis added.>> The OP is 13! :rolleyes:
I've edited it to remove the offending word!
 
  • Haha
Likes TensorCalculus
  • #91
Mozibur Rahman Ullah said:
I've edited it to remove the offending word!
It doesn't matter whether it's "stage of career" or "stage of education". The fact remains the OP is 13. Responses should be tailored accordingly.
 
  • Like
Likes TensorCalculus
  • #92
Hey, @Mozibur Rahman Ullah - thank you so, so much for putting in all of the time and effort to explain that to me (and sorry for the late response) - it means a lot to know that there are people willing to spend so long helping out a curious 13-year-old girl :D
As for your explanation, I thought it was great! I had to read it through a few times to fully get what was going on, but most of it was really understandable - I got what you were saying, and the only searches I made were on the lines of "what does the word auxiliary mean", or "what is the tensor product" so nothing really that was instrumental to your approach. I enjoyed it a lot - and whilst I may be (very) young, I think you dumbed it down very well :D.
a few things:
Mozibur Rahman Ullah said:
The map T is called the tangent functor and this has nice properties. For example, if M & N are manifolds then T(M×N)=TM×TN
Ok so this is absolutely not the correct thing to ask, especially because I obviously don't really know the appropriate maths but my curiosity is really getting the better of me - why? For me it just doesn't seem immediately intuitive. (also, is the multiplication of manifolds similar to to taking the cross product of vectors? It seems that way... ish)
Mozibur Rahman Ullah said:
t's best to see this as an illustration but I don't know if Physics Forums supports images.
It does!
Mozibur Rahman Ullah said:
Its also worth mentioning that this rule is basically the chain rule we learn to love in calculus, generalised to multivariable calculus and then to manifolds. The way to show this is to work locally, that is by fixing charts and then working out the various expressions.
Yeah, this cleared up a lot. In my mind, it's basically "generalised chain rule" now - which actually makes it very easy to kind of get a gist of what's going on :D
Mozibur Rahman Ullah said:
It's been a long post but I hope this gives you some sense on how the exterior covariant derivative comes about. To be honest, I don't expect you to grasp all this as I've skipped over a lot of detail - for example, what is the wedge product. And don't feel frustrated if it escapes you, as I've already said the exterior covariant derivative is really, really advanced stuff. But I do hope what I've written gives you a feeling for the shape of the terrain you would have to cover to rigourously learn this stuff. If you have any questions on what I've written then feel free to ask.
It's been long - but actually great! I understood where it came from (though I have to agree with @CrysPhys that I have a long way to go till I'm ready to study it in any depth whatsoever) and it was really, really interesting. I think I would have been frustrated if it slipped past me (though most of it didn't... thankfully), because now you've made me curious - and once I'm curious there's no stopping me :) Thank you for the explanation once again - it was really eye-opening!
CrysPhys said:
<<Emphasis added.>> The OP is 13! :rolleyes:
yeah.... 13 but curious! :) (though thanks for the reminder, once again my curiosity gets the better of me sometimes, honestly without you I'd already have confused myself 50 times over via random sources on the internet about differential geometry because @Mozibur Rahman Ullah got me so excited) You were also right about the university courses - I've almost finished the calculus one, and it was very useful - I learnt a lot!
CrysPhys said:
It doesn't matter whether it's "stage of career" or "stage of education". The fact remains the OP is 13. Responses should be tailored accordingly.
I thought it was tailored pretty well! Extremely A bit ahead of what I'm doing right now, but interesting and understandable (even if I in reality know nothing about the subject and probably don't even really understand the exterior covariant derivative at all it just feels like it).
 
  • #93
Mozibur Rahman Ullah said:
Great, I'm glad you feel its worth learning. But I'd hold your horses about learning this stuff seriously at this stage of your education. This is advanced stuff and the exterior covariant derivative is really advanced stuff! Typically, you will learn vector analysis in undergraduate courses and differential geometry in graduate courses and the exterior covariant derivative is most likely taught at a second course at this level. Certainly I wasn't introduced to it in my masters course on theoretical physics. Nor does the textbook I recommended, Lee's Smooth Manifolds explains this. I first learnt about it in Baez & Muniain's Gauge Fields, Knots and Gravity which I have already recommended. It takes the physics approach to differential geometry and so avoids all the very careful constructions in math whilst still being relatively rigorous. The chapter you should look at is chapter II - 3, Curvature and the Yang-Mills equation, pg.250. But I urge you to study the text systematically, or at least go through it trying to understand the concepts that they introduce to get a sense of the terrain. It pays to go through the text step by step. But I also think at this stage of your education you should be getting the fundamentals of physics down - that is really, really important - and not be distracted by advanced and really advanced stuff. I promise you they will still be waiting for you later on in your career. It's a problem that I've had in my own education, so I feel your frustrations. I explained what I did in my previous post as motivation that this stuff is worth learning and not as an indication that you should be learning this stuff at this stage of your education.

Nevertheless, I've written an exposition below which I hope will give you a sense of the terrain without getting bogged down in detail. I hope it helps without further confusing you and leading you astray!

First, to really get grips with the exterior covariant derivative you need to be comfortable with manifolds, tangent bundles and vector bundles as well as their spaces of sections - basically tangent fields and vector fields, as well as the wedge product (a generalisation of the cross product which works for all dimensions, the cross product only works in 3d) and tensor product and of course differential forms and the exterior derivative, oh as well as understanding that tangent fields can also be interpreted as certain differential operators. This is a long list of mathematical technology, so I've written an exposition that gives you a birds-eye view of some of the neccessary concepts without getting into the detail that's needed to really grasp these ideas and make use of them.


The first thing to learn is manifolds. This is basically a formalisation of one aspect of the notion of covariance. In fact Einstein said:


"So there is nothing for it but to regard all imaginable systems of coordinates, on principle, as equally suitable for description of nature"


Manifolds are spaces that are equipped with charts, that is coordinate systems. They are also equipped with transition functions that map between these different charts. This sounds complicated, but it's not so bad as we learn to manipulate manifolds. For example, if you had two manifolds ##M## & ##N## then you can multiply them ##M \times N## and you can take their disjoint sum ##M \sqcup N##. For example if we took the real line is a manifold as well as the circle. If we multiply them we get the infinite cylinder and if we add them we just get the line 'next' to the circle. So we have a calculus of manifolds. It's also important to distinguish between manifolds by themselves and manifolds that are embedded in Euclidean space. For example, if you imagine a sphere - by the way, mathematicians call the surface of a ball, a sphere and the interior of a ball, is just called a ball, then you will probably imagine a sphere in space. This makes it easy to see what the tangent planes on a sphere are. This is a sphere embedded in space. When a mathematician refers to a sphere, it is a sphere all by itself, sort of hanging in the void. This makes it difficult to understand what the tangent planes are as there's nowhere we can take the tangent plane. There's only the void surrounding the sphere. However, there is a way of introducing the tangent planes. This relies on maths but a quick way of seeing this, is simply to invoke a theorem that all manifolds are embeddable in a Euclidean space of high enough dimension and then just take the tangent planes as usual. The manifold with all the tangent spaces attached is called the tangent bundle over that manifold.


Now if ##M## is a manifold - for example a sphere - then ##TM## is the usual notation for the tangent bundle over ##M##. The map ##T## is called the tangent functor and this has nice properties. For example, if ##M## & ##N## are manifolds then ##T(M \times N) = TM \times TN## and obviously ##T(M \sqcup N) = TM \sqcup TN##. This is already important. For example, the tangent space to the infinite cylinder can now be easily calculated. It is ##T(S^1 \times \mathbb{R}) = T(S^1) \times T(\mathbb{R})##.


It also turns out that if we have maps ##f: M \rightarrow N## and another map ##g: N \rightarrow P##, then ##T(g\circ f) = Tg \circ Tf##. To understand this intuitively, it best to see the situation geometrically. When we map one manifold to another, then we also map the tangent planes on the first manifold to the second manifold. Then if we have a composition of maps then the tangent maps also compose. It's best to see this as an illustration but I don't know if Physics Forums supports images. Its also worth mentioning that this rule is basically the chain rule we learn to love in calculus, generalised to multivariable calculus and then to manifolds. The way to show this is to work locally, that is by fixing charts and then working out the various expressions.


Now the covariant derivative ##D## is a replacement for the directional derivative in differential geometry. The usual notion of a directional derivative doesn't work as it relies on the embedding of the manifold in Euclidean space. However, that theorem of embedding the manifold in a Euclidean space of high enough dimension comes in useful again. It turns out that the covariant derivative is the orthogonal projection of the directional derivative onto the tangent space. I say tangent space rather than tangent plane because whilst a 2d manifold has a 2d tangent space - aka a tangent plane - a n-dimensional manifold has a n-dimensional tangent space. Usually, when the covariant derivative is introduced it is done intrinsically but I find this geometric way of defining it more intuitive.


We need the covariant derivative because the exterior covariant derivative combines the exterior derivative and the covariant derivative. I haven't introduced the exterior derivative but we can at least see how the exterior covariant derivative is defined on Baez & Muniain, pg.250


We are going to go for an inductive definition. This is because we have differential 0-forms, 1-forms, 2-forms all the way upto ##n##-forms where ##n## is the dimesion of the manifolds. A differential k-form is said to have order ##k##. The space of k-forms on a manifold ##M## is denoted by ##\Omega^k(M)##. It will turn out that the exterior derivative of a k-form will result in a form of one degree higher, aka a k+1-form. In symbols:


##d: \Omega^k(M) \rightarrow \Omega^{k+1}(M)##


We really ought to have a superscript ##k## on the ##d## but this is just usually left implicit to avoid notational clutter.


Now, its important to realise 0-forms on a manifold are precisely real valued functions on the manifold. Then the exterior derivative ##d## of a differential 0- form - aka a function ##f## - is defined by:


##df(v) := v(f)##


Here ##v## is a vector field, really a tangent field. Now I've already said above that tangent fields can be reinterpreted as a certain differential operator. So ##v(f)## is saying ##v## is deriving ##f##.


Now the exterior covariant derivative is defined by introducing an auxilary covariant derivative ##D## and this must live on some vector bundle ##E##. We haven't introduced vector bundles before. They are a generalisation of tangent bundles. And just as tangent bundles have a covariant derivative, so do vector bundles. Now instead of having k-forms we have instead E-valued k-forms. The space of all E-valued k-forms is written as ##\Omega^k(M, E)##. Then the exterior covariant derivative maps E-valued k-forms to E-valued k+1-forms. Or in symbols:


##d^D : \Omega^k(M, E) \rightarrow \Omega^{k+1}(M, E)##


Here again the ##d^D## ought to have a superscript ##k##, but its left implicit. Now to run the inductive definition of ##d^D## we need to find out what E-valued 0-forms are. These are just vector fields of E. We denote them by latin letters, so in the following formula for ##d^D##, by the letter s.


##(d^Ds)(v) = D_vs##


Thus we see in the zeroth step of the inductive definition we see we have replaced the derivative ##v## acting on ##f## by ##D_v## on ##s##. Here ##f## is a 0-form, aka a function, and ##s## is an E-valued 0-form, aka a vector field in ##E##.


We then inductively define ##d## for higher differential forms via a variation of the Liebniz rule for the derivative of products (Baez & Muniain, pg.63) :


##d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta##


Here ##k## is the order of the differential form ##\alpha##. The symbol ##\wedge## is the wedge or exterior product - it's the generalisation of the cross product to any dimension because the usual cross product just works in 3d only. The corresponding inductive definition for the exterior covariant derivative is on Baez & Muniain, pg.250. It is also a variation on the Liebniz product rule:


##d^D(s \otimes \alpha) = d^Ds \wedge \alpha + s \otimes d\alpha##


Here ##s## is an E-valued 0-form and ##\alpha## is an E-valued k-form. Here the symbol ##\otimes## is the tensor product.


It's been a long post but I hope this gives you some sense on how the exterior covariant derivative comes about. To be honest, I don't expect you to grasp all this as I've skipped over a lot of detail - for example, what is the wedge product. And don't feel frustrated if it escapes you, as I've already said the exterior covariant derivative is really, really advanced stuff. But I do hope what I've written gives you a feeling for the shape of the terrain you would have to cover to rigourously learn this stuff. If you have any questions on what I've written then feel free to ask.
Nitpick : ## T(M \times N) ##~##T(M) \oplus T(N)##.
 
  • Like
Likes TensorCalculus
  • #94
TensorCalculus said:
>thank you so, so much for putting in all of the time and effort to explain that to me (and sorry for the late response) - it means a lot to know that there are people willing to spend so long helping out a curious 13-year-old girl :D

Ah, I thought you were a 13 old year boy! A gender assumption on my part! I'm glad that what I wrote was taken in by you and understood. Richard Feynman once said that you only understand a concept if you can explain it to a student - so I guess I can say I understand the exterior covariant derivative! ;-).
TensorCalculus said:
>what is the tensor product?

The tensor product is taught badly. One reason is that they don't use geometrical intuition and its usually defined in a very mathematical way which is difficult to understand unless you've done the relevant math - basically the free vector space on a basis and a quotient of a vector space.

The tensor product of two vectors results in a higher dimensional vector. There are actually two ways to think of higher dimensional vectors. The simplest way is that the vector remains the same - an arrow - but it now lives in a higher dimensional space. This isn't the tensor product. The second way of thinking about it is that the arrow itself becomes higher dimensional. This is the tensor product. Thus a 3d arrow is the tensor product of 3 vectors ##u \otimes v \otimes w##. Whereas an ordinary vector has only one way to add, a 3d arrow has three ways to add. That's not so scary when you see it visually - however I haven't figured out how to insert diagrams here yet. A 2d arrow is ##u \otimes v## and is visually represented by a parallelogram in the obvious way. However, there is a proviso - you can internally rescale it without changing the tesor. By internal rescaling I mean you can rescale one side, say ##u##, by a factor of ##a## whilst rescaling the other side, ##v## by ##1/a##. This is because the magnitude of a tensor is its area. For a 3d tensor, which will be a parallelepipid, its magnitude will be its volume. And so on. Another thing to note is that in this geometric picture you can only define tensors upto the dimension of the ambient space. But in fact, it can be defined for arbitrary dimension. This is best shown mathematically.
TensorCalculus said:
>also, is the multiplication of manifolds similar to to taking the cross product of vectors? It seems that way... ish
the multiplication of manifolds follows the Cartesian product of sets. Have you come across this? Basically, a manifold is a set with a smooth structure. Here, the smooth structure is the set of all charts on the manifold. When you multiply two manifolds ##M##, ##N## together you first multiply the two sets together to get ##M \times N## and then the smooth structures on ##M## and ##N## determines a smooth structure on ##M \times N##. We still use the symbol for the Cartesian product ##\times## for the product of manifolds.

I think you also asked why ##T(M \times N) = TM \times TN##. The best thing would be to try a few low dimensional cases to see why its true. For example, the plane which is ##\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}##, the torus ##T^2 = S^1 \times S^1##. Recall here that ##S^n## is the n-dimensional sphere, and so ##S^1## is just the circle. Whilst ##T^2## is the torus which is 2d. And the infinite cylinder ##Cyl^2 = \mathbb{R} \times S^1##.
TensorCalculus said:
>It's been long - but actually great! I understood where it came from @CrysPhys that I have a long way to go till I'm ready to study it in any depth whatsoever) and it was really, really interesting. I think I would have been frustrated if it slipped past me (though most of it didn't... thankfully), because now you've made me curious - and once I'm curious there's no stopping me :) Thank you for the explanation once again - it was really eye-opening!

You're welcome. I think it's fantastic that a thirteen year old is taking so much interest in advanced ideas. You've done remarkably well in taking it in. Well done! If you have any further questions on the material, you're welcome to ask them.
 
  • Informative
Likes TensorCalculus
  • #95
Mozibur Rahman Ullah said:
Whereas an ordinary vector has only one way to add, a 3d arrow has three ways to add. That's not so scary when you see it visually - however I haven't figured out how to insert diagrams here yet. A 2d arrow is u⊗v and is visually represented by a parallelogram in the obvious way
Aren't you talking about multivectors here?
 
  • Like
Likes TensorCalculus
  • #96
Hey y'all... we do have a differential geometry forum here... hint hint. :wink:
 
  • Haha
Likes TensorCalculus
  • #97
Muu9 said:
Aren't you talking about multivectors here?
Not quite. Usually k-multivectors (or k-polyvectors or k-vectors) are wedge products of k vectors rather than the tensor product of k vectors. But given that there is no specialised term for them, I think it's no great stretch to use the same term for them.
 
  • Like
Likes TensorCalculus
  • #98
jtbell said:
Hey y'all... we do have a differential geometry forum here... hint hint. :wink:
I'm afraid it's a lost cause. :oldbiggrin:
 
  • Haha
Likes TensorCalculus
  • #99
jtbell said:
Hey y'all... we do have a differential geometry forum here... hint hint. :wink:
I've noticed. But this thread began as a reply to a request for advice on self-studying physics beyond high school physics and I mentioned differential geometry and it went downhill from there ;-).
 
  • Haha
Likes TensorCalculus
  • #100
Classic case of "topic drift." :cool:
 
  • Haha
Likes TensorCalculus

Similar threads

Replies
16
Views
2K
Replies
71
Views
648
Replies
10
Views
5K
Replies
22
Views
6K
Replies
7
Views
2K
Replies
7
Views
3K
Back
Top