Mathematic subjects to self study.

[FuturePhysicist]

I am currently in high school and have been driven to become a theoretical physicist in an unsure field for quite some time now. I have been unfairly placed in a very easy math class even though I already know it. I have been studying upon subjects such as algebra 2 and pre-cal and recently got a book on calculus. I am quite a fast learner, so it won't take me long to finish the subject. I'm curious of what subjects I should study (in order) to become successful in physics. I have already compiled a small list that is not complete and that I am unsure of. Cal 2, Cal 3, advanced calculus, ordinary differential equations, linear algebra, and real analysis. I would like to see a list (in order) of all of the courses that I should study after these. And I would like to know if there is a mistake in my list.

 

[axmls]

Probably the best thing you can do with your time is to reinforce your knowledge of the basics--problems with the fundamentals of algebra/trig are the things that come back to haunt a lot of students. This is especially since you'll probably have to take all of those courses in college anyway. But if you do need a list of things physicists should know, you should be very comfortable with differential calculus, then move on to integral calculus, then multivariable calculus. After that, you can learn about ordinary differential equations. Then perhaps vector calculus and partial differential equations. To be honest, linear algebra should be possible before even learning calculus, and it could serve to enhance the education you already have in algebra. You could always look into that. I believe it's quite important for QM. Upper-level math courses like real analysis, topology, etc. aren't things you should be worried about before you've taken your calculus courses.

Basically what I mean is, you will study all of those subjects if you major in physics. Don't try to rush into those, because math isn't like history. If you skip learning about the 1600s, then you can still probably do well learning about the 1700s, but in math, if your fundamentals aren't strong, then you're likely to struggle with more advanced topics.

 

[MidgetDwarf]

Revise pre calcUlu material, then start learning calculus.

For calculus book recommendations, I recommend stewart calculus(5th edition and under) used with thomas calculus with analytic geometry 3rd edition. I would work from both at the same time. For a person new to calculus Thomas does not give an easy readable introduction to the chain rule, epsilon delta concept of limits, and there's one major topic I'm missing. However, Thomas explains the concept of derivatives/iintegration, work, optimization, and mean/rolls theorem extremely thorough.

Proofs of the theorems are explained with great detail in thomas, which leads to better understanding of the calculus.

the books balance each others faults perfectly.

 

[MidgetDwarf]

Probably the best thing you can do with your time is to reinforce your knowledge of the basics--problems with the fundamentals of algebra/trig are the things that come back to haunt a lot of students. This is especially since you'll probably have to take all of those courses in college anyway. But if you do need a list of things physicists should know, you should be very comfortable with differential calculus, then move on to integral calculus, then multivariable calculus. After that, you can learn about ordinary differential equations. Then perhaps vector calculus and partial differential equations. To be honest, linear algebra should be possible before even learning calculus, and it could serve to enhance the education you already have in algebra. You could always look into that. I believe it's quite important for QM. Upper-level math courses like real analysis, topology, etc. aren't things you should be worried about before you've taken your calculus courses.


Basically what I mean is, you will study all of those subjects if you major in physics. Don't try to rush into those, because math isn't like history. If you skip learning about the 1600s, then you can still probably do well learning about the 1700s, but in math, if your fundamentals aren't strong, then you're likely to struggle with more advanced topics.

I would advice from learning linear algebra before calculus 1 or even calculus 2. My reasons are as follows:

Sure linear algebra has hardly any calculus in it (a few problems here in there that can be avoided). TO do linear algebra basic arithmetic properties are all a person needs computational y. However, the theory can become abstract and lack of mathematical maturity can lead to miss understanding or overlooking important yet miniscule details.

to learn linear algebra mathematical maturity is a must.

For linear algebra. I would recommend Paul Shields Linear Algebra (basic introduction but a goody), Serge Lang Introduction to Linear algebra, or replace Shields with Anton Elementary Linear Algebra book.

 

[QuantumCurt]

I wouldn't have wanted to take linear algebra prior to taking at least calculus 1 and calculus 2. It's true that linear algebra often doesn't involve calculus at all, but linear algebra can be pretty abstract if one hasn't already developed a bit in mathematical maturity.

I'd second the suggestion for Lang's book. Make sure to get the Introduction to Linear Algebra book though. The one that's titled just "Linear Algebra" is a much higher level book than the introduction to linear algebra book. Gilbert Strang has several really good linear algebra texts as well. This book accompanies his lectures on MIT OpenCourseWare. https://www.amazon.com/dp/0980232716/?tag=pfamazon01-20

 

[MidgetDwarf]

Thanks for the correct quantom. I meant to type, I would advise not to. my tablet has a nasty habit of auto correcting my typing.

 

[MidgetDwarf]

I wouldn't have wanted to take linear algebra prior to taking at least calculus 1 and calculus 2. It's true that linear algebra often doesn't involve calculus at all, but linear algebra can be pretty abstract if one hasn't already developed a bit in mathematical maturity.

I'd second the suggestion for Lang's book. Make sure to get the Introduction to Linear Algebra book though. The one that's titled just "Linear Algebra" is a much higher level book than the introduction to linear algebra book. Gilbert Strang has several really good linear algebra texts as well. This book accompanies his lectures on MIT OpenCourseWare. https://www.amazon.com/dp/0980232716/?tag=pfamazon01-20

However, I would disagree with Strangs book. The book lacks rigor, strang explanations are extremely wordly, and the book seemed liked a random assortment of matrix operations. I would strongly advise NOT to use Strangs book. Lang does a better written and more rigorous approach than Strang. Also, Lang has a habit of unifying the theory, so the reader knows the why. However, Lang suffers a lack of problem sets, so Anton or Paul Shields is a must if using Lang.

Only problem with shields is that it goes up to at most 3-ddimensions and does not handle n-sno space if I recal. However, Lang book does treat n-space so this problem is remedied.

 

[IGU]

This question has been answered many times. This:
http://www.staff.science.uu.nl/~Gadda001/goodtheorist/index.html
Just follow Gerard 't Hooft's guidance and you'll be in good hands.

 

[Arsenic&Lace]

Advanced calculus and real analysis are completely useless for theoretical physics and perhaps even harmful. I do enjoy them as brain teasers though, so explore real analysis for fun, provided you bear in mind that none of it is relevant to physics. However you'll gain enormously more per unit time learning how to program, which is a vital skill for any theoretical physicist since all modern physics problems are intractable without computers.

 

[axmls]

Advanced calculus and real analysis are completely useless for theoretical physics and perhaps even harmful. I do enjoy them as brain teasers though, so explore real analysis for fun, provided you bear in mind that none of it is relevant to physics. However you'll gain enormously more per unit time learning how to program, which is a vital skill for any theoretical physicist since all modern physics problems are intractable without computers.

I can't believe I forgot to mention this. If there's anything productive the OP can do with their time, it's learning how to program. They'll learn whatever math they need in college, but the ability to program is something that can always be improved outside of school.

OP, I recommend you start learning some Python. It's a great introductory language, and either Python or some other language (which Python will help you learn) will be absolutely invaluable to work in physics.

 

[Arsenic&Lace]

Python is becoming increasingly ubiquitous too, so while you may need C or C++ at some point you may be able to do quite a bit of work in python anyway.

 

[micromass]

Advanced calculus and real analysis are completely useless for theoretical physics and perhaps even harmful. I do enjoy them as brain teasers though, so explore real analysis for fun, provided you bear in mind that none of it is relevant to physics. However you'll gain enormously more per unit time learning how to program, which is a vital skill for any theoretical physicist since all modern physics problems are intractable without computers.

To the OP: please keep in mind that Arsenic&Lace is a math-hater. His opinion is a minority, so please take it with a grain of salt.

Real analysis is certainly not harmful (no correct knowledge is harmful). Whether it is useful or not depends a lot on what you plan to do. For the ordinary physicist, it certainly is not very useful. However, if you're going into more theoretical physics (like theoretical Relativity), then you can't really afford not to know real analysis (and specifically: differential geometry, topology and manifolds).

 

[Arsenic&Lace]

To the OP: please keep in mind that Arsenic&Lace is a math-hater. His opinion is a minority, so please take it with a grain of salt.

Real analysis is certainly not harmful (no correct knowledge is harmful). Whether it is useful or not depends a lot on what you plan to do. For the ordinary physicist, it certainly is not very useful. However, if you're going into more theoretical physics (like theoretical Relativity), then you can't really afford not to know real analysis (and specifically: differential geometry, topology and manifolds).

I take umbrage at being called a math hater, since I'm quite fond of math. Can't we all just get along?

My opinion is certainly not in the minority. A friend of mine with a background in electrical engineering and pure mathematics is studying control systems, the only subject in engineering I am aware of which seems to use quite a bit of pure mathematics. Imagine his surprise, when after he'd spent several semesters in classes like real analysis and the differential topology and geometry of manifolds in control systems that when he got into industry to work for Raytheon, they bluntly told him that the only things they cared about was an understanding of how to write massively parallelized Monte Carlo codes! A lack of interest in pure mathematics is wide spread in applied mathematics disciplines, because most of these disciplines have been pushing the boundaries of complex systems, even in more pure disciplines such as condensed matter physics, and it turns out that when you want to figure out whether or not your autonomous missile will guide itself in high turbulence or how that protein will fold, pure mathematics has rarely proven itself to be particularly useful.

Pure math courses can be harmful to a physicist if he internalizes the culture of pure mathematics, which as its name implies is a culture suitable for doing pure mathematics, and is a pretty dreadful culture for doing physics. The perspective taught in real analysis on series is really nonsensical in the context of quantum field theory and vice versa for instance. Of course if you separate the two disciplines properly pure math serves as harmless fun.

Every theoretical physicist I've spoken to except a string theorist has told me that they never took pure math courses, and I'm not sure string theorists are really qualified to be described as theoretical physicists anyway (Witten strikes me as an individual who conflates pure mathematics and physics; did anybody read the IPMU interview where he crooned about how he couldn't wait for number theory to be found to have fundamental significance to theoretical physics, as if this were inevitable?). So I'm not sure where micromass's claim of requiring knowledge of real analysis to work in relativity is really true. I suppose if you like pontificating ala Hawking or Penrose you might need a bit of pure math, but if you're interested in actually connecting with experiments (i.e. actually doing what a theoretical physicist is supposed to) you'll find real analysis completely irrelevant, although at that point you probably won't be working in theoretical relativity.