Mathematical rigor in physics (calculus)

>>8559804
it depends, you'll need to know some math eventually, but it's better to learn it as you go, whenever you need it. you'll need to get up to differential geometry to understand relativity well, but there are good "for physicists" resources available

>>8559804
It depends on what your goal is. If you want to eventually study high level theoretical physics, you might as well start doing the rigorous mathematics now.

>>8559814
Where do I start?

>>8559822
1) Go to college

>>8559826
>paying $3000+ for Calculus

>>8559826
Are you implying autodidactism is not as effective as formal education?

>>8559814
I agree. If you just care about undergrad, then don't sweat it. If you are considering grad school, take some analysis (real and complex), abstract algebra, and really any course requiring rigorous proofs. A lot of what you learn you will never apply to physics, but you'd be surprised how helpful it is to have this knowledge on hand. Also just practicing thinking like a mathematician helps. To go through physics just believing things because you were taught them is very unfulfilling in my opinion.

>>8559833
yes, obviously

>>8559848
Why do you think that? Not trying to be confrontational. I'm honestly curious and want to see what makes a formal education better than just reading a textbook and doing the problems.

>>8559804
Isn't that the guy who jumped off a bridge because he couldn't find employment with a PhD in physics? Lol.

>>8559850
the ideas are very valuable, and they need a lot of time to mature on your own.

as a mathematician, I can read a long winded proof, and understand all the steps, and confirm the theorem is true, and understand what it says, but learn absolutely nothing. conversely, a professor can draw a very simple, small picture and move his arms a bit and suddenly the whole significance of the theorem is clear, the proof is trivial, and it falls into the context perfectly. math is about developing the correct intuition, and learning from people who have developed it for years helps a lot.

>>8559860
There exist online lectures and an abundance of online resources to help develop intuition.

>>8559864
there aren't many online lectures in math beyond the basic materials, and what online resources can help you develop intuition? conversation with other students and professors is what develops intuition

>>8559804
>Is it necessary to learn calculus rigorously (with proofs) to have a good understanding of physics

Yes because you stand little change of understanding advanced methods without it.

>>8559867
Various forums and shitpost centers (4chan) where mathematics graduates congregate. Also, I don't know what you mean by "basic materials", but I'd wager that there is at least a complete undergraduate course uploaded by a reputable university somewhere.

>>8559872
the thing is, you definitely could if you're very motivated. but going to uni makes it far easier, and that's the whole point i'm trying to make. if you're very hardworking, talented and motivated, you're going to make it anywhere, anyway

>>8559850
You can barely get past sophomore level topics with the quality channels. Nothing can really replace learning complicated shit from someone who knows the topic irl, especially at a research university.

>>8559898
Ok, but what do I do if I can't attend a university?

>>8559901
why can't you? community colleges are affordable

>>8559850
Maybe 1 in a million people has the discipline to actually go through with it.

Most likely you'll be studying some stuff on your own, you'll think you understand it, but you really don't. You'll also most likely skip over huge parts that are very boring but very important.

I have this all the time when in summer vacations I try to learn some stuff in advance, and then when we go over that same topic in the lecture a couple of weeks later, I realize I actually didn't understand it at all.

>>8559852
Yes
http://www.dailymail.co.uk/news/article-2304096/Graduate-physics-PhD-31-fell-death-block-flats-taking-job-centre-qualified-for.html

>>8559804
What kind of physics do you want to do?

If you wanna do theoretical, you'd better sit yo ass down and read up some advanced, rigorous math.

If you're just gonna do engineering or something, then rigor isn't very important.

>>8559850

I don't think there's any reason autodidactiveness can't be as effective as formal education, just for learning purposes. However, most often this isn't the case.

With a formal education, you get access to far more resources, and professors who help and peers who learn with and from you. Of course, the amount of benefit from these things varies among different people.

>>8559901
You're pretty much shit out of luck unless you're some kind of Newtonian boy wonder then. Research today requires access to lab technology (which is often very, very expensive) and correspondence with equally-minded individuals, both of which you aren't likely to acquire without a degree.

If you just want to understand what's already known, go for it. There's no reason you can't learn undergrad physics on your own. If you want to conduct research, go to university.

>>8560023
You can start with MIT ocw. I'm not sure the courses offered there cover all of undergrad physics, but you can at least get up to speed on the maths you'll need (calc 1-3, linear algebra, diff.eq, and complex analysis, and go through the basics of mechanics, etc.

I have some relevant experience, I think.
I used to study a degree that combined physics and another major. The physics sufficed, it was applied to everyday situations, and mostly it was just working with formulas.

Yet, I was pretty unsatisfied. I felt like I lacked the fundamental knowledge, and although I strongly believe that my physics was better than that of a chem student, I was still way below a physics student. It's not like phys is actually that much harder, but the fundamental thought processes are something that you should have practised, in order to get good at it.

So yeah, I'd learn the fundaments first. These can be learned pretty quick, but really don't just stop after calc 3. Take three to four courses that apply the calc knowledge to practise.

>>8559804
>rigorously (with proofs)
In a word, No. It is just a waste of your time. You need a good feel for the math but anal-retentive proofs add no value to the physicist.

Instead spend you time learning physics and enough math to understand the physics.

>>8559994
>advanced, rigorous math.
Advanced yes, super-rigorous no.

>>8559994
This
/thread.

>>8560070
Also large swathes of math are useless for physics (at least until you get to the frontiers and maybe not even then). They will try to make you learn e.g.abstract algebra including the Sylow theorems and Galois theory but most of it is not relevant.On the other hand group representations are quite useful and often are not covered in a standard AA course.

>>8559804
It is if you don't want to create b.s. mathematical proofs to substantiate your claims in your physics paper. If you are talking about applied physics, then don't worry about the math, you can just use the formulas in the power points.

>>8559804
Honestly as a physicist you can sit in a real analysis course, which would be your basic starting point to get into math, and still wonder where you're going to apply all that shit.

>>8559804
If you plan on doing any theoretical physics then yes. Just about any rigorous math you can learn will be useful. Even the most advanced stuff like functional analysis, algebraic geometry, differential geometry, algebraic topology can and is applied in modern theoretical physics. Learning to prove things is necessary to even make use of the high powered stuff. Even highly abstract stuff like categories are used and it's not really feasible to work with them and not prove things rigorously.

If you don't want to do theoretical physics then no. Not at all.

>>8559833
Despite what he says self-teaching is 100x better.

>>8560212
Not at all, being at a good school is 100x better

>>8560215
school is self-teaching but it costs money

>>8559867
So, how ramanujan did it alone? there must be a method.

>>8560215
I heard that schools are there to teach you how to self-teach, so....

>>8560695
yeah it's called not being a retard who needs a teacher to baby them

Anybody in this thread that says that rigorous math is not absolutely mandatory for physicists are fucking retards and have no idea how complex problems work, I'll give you a very simple example why is it very important to understand and check theorem assumptions

Take a trivial example: the length of the curve of the trajectory of a projectile (standard physics I problem).

If you for example work out the integral and get an expression that outputs the length of the trajectory based on the initial angle, you might for example want to check your solution for the limit cases [math] \alpha = \frac{\pi}{2}, \alpha = 0 [/math] and since the expression isn't really trivial, you might have to use L'Hospital rule for example. You will have to re-arrange your equation to fulfill the assumptions, that is the limit will be equal to an indefinite expression [math] "\frac{0}{0}" , "\frac{anything}{\infty}" [/math]. If you however re-arrange the expression in a way that your limit gives anything else than these cases, your L'Hospital use will give wrong solutions.

There are other ways to solve this problem, however sometimes when you need a general solution for a problem, you might come into a situation like this and to recognize which solutions your model gives are correct, you need to understand when the theorems you are using work.

Anybody who says otherwise never worked with PDEs or dynamical systems generally.

>>8560750
What would be the better learning order? Math first, physics first, or the two in tandem, learning the necessary math rigorously when it is needed for a physics concept?

>>8559835
I second this notion

>>8560921
I would probably do math first and then catch up with the physics at relevant points, I feel that way you gain a strong understanding by understanding the mathematics intuitively yet and mastering it in such a way that you can apply it to physical problems.

>>8560750
>attempts to argue that physicists need advanced math
>all he can come up with is a trivial application of calculus 1 which requires no proof or theory
you're retarded

>>8560921
don't listen to that idiot, other people already gave you some useful perspectives

>>8560921
learn it as you go bro. >>8559835 thirded, take a couple of core electives

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>>8560921
>What would be the better learning order?
Learn in parallel - learn the math as you need it. Early on I tried to proactively learn the math but I found I wasted a lot of time learning stuff I didn't need and will never need e,g. rings, ideals, most of group theory.

Learn as needed.

Also people will tell you that you cannot learn on your own and college is better. I have done both and my conclusion is that learning on your own is better as long as you have the discipline and !!!do the exercises!!! and redo the proofs in your own words and diagrams.

Going to college you tend to rush things and cram for exams and then forget it. I did linear algebra at college and forgot all of it. Now having studied it myself it is ingrained deeply.

The only case where a college would be better is if the college is genuinely first tier top 10. Even then there is little contact with the actual top profs before grad school, if then.

If you want to self-study I suggest two books "A mind for numbers" and "deep work". They might change your whole experience.

>>8559804
no.
Math is only a tool, a carpenter doesn't need to know how a nail is made, to use it.

I'm not saying it would be totally useless knowledge, but you will spend your time better learning calculus the american way, and in the future, if you have spare time, pick a Spivak/Courant kind of book.

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>>8561788
>the american way

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>>8563495
My dude