Rigorously covering mathematical prerequisites for Graduate courses

[crimeanwarcrimes]

I am a high school who will be graduating next year. I am almost done with Purcell and Morin which I worked through in my past holidays. However, I am still not proficient in various topics like Linear Algebra, Basic Analysis, and I quite often encounter various new things that I haven't even heard of. I am looking for any guidance to systematically cover mathematics intuitively and rigorously over the span of 1-2 months, apt enough to cover most of the Mathematical prerequisites for graduate courses. Keep in mind that I will be having the whole for myself which I will spend doing physics, for the most part, Any help will be appreciated

 

[S.G. Janssens]

Sorry, but I am a bit confused. You write you are in (?) high school, but you are looking to prepare for graduate physics courses? Isn't there normally an undergraduate physics program in between that also comes with appropriate mathematics courses? Then, after that, you proceed with a graduate physics program.

The mathematical prerequisites for undergraduate physics (which comes after high school) are lighter than the prerequisites for graduate physics (which comes after undergraduate physics).

Maybe it helps if you give some examples of physics courses (with descriptions) that you want to be prepared for.

 

[crimeanwarcrimes]

Sorry, but I am a bit confused. You write you are in (?) high school, but you are looking to prepare for graduate physics courses? Isn't there normally an undergraduate physics program in between that also comes with appropriate mathematics courses? Then, after that, you proceed with a graduate physics program.

The mathematical prerequisites for undergraduate physics (which comes after high school) are lighter than the prerequisites for graduate physics (which comes after undergraduate physics).

Maybe it helps if you give some examples of physics courses (with descriptions) that you want to be prepared for.

Well I am in high school but I wanted to cover my bases by first strengthening my mathematical skills before I moved on to some calculation heavy stuff

 

[symbolipoint]

Well I am in high school but I wanted to cover my bases by first strengthening my mathematical skills before I moved on to some calculation heavy stuff

As SOON as you begin college or university, be sure to go through a sequence of courses in THIS way, unless admissions testing determines exactly where in the sequence you should start:

1. Introductory or Elementary Algebra
2. Intermediate Algebra
3. Geometry
4. Trigonometry OR Elementary Functions
5. Possibly one separate one-semester course on Trigonometry
6. Calculus & Analytical Geometry, I, II, III, and a possible IV.
7. Introductory Differential Equations & Linear Algebra (a combination course, usually "terminal").

Some variations among the list are possible depending on institutional organization.

 

[crimeanwarcrimes]

As SOON as you begin college or university, be sure to go through a sequence of courses in THIS way, unless admissions testing determines exactly where in the sequence you should start:

1. Introductory or Elementary Algebra
2. Intermediate Algebra
3. Geometry
4. Trigonometry OR Elementary Functions
5. Possibly one separate one-semester course on Trigonometry
6. Calculus & Analytical Geometry, I, II, III, and a possible IV.
7. Introductory Differential Equations & Linear Algebra (a combination course, usually "terminal").

Some variations among the list are possible depending on institutional organization.

I have completed almost all of it and few things beyond that like Vector Calculus ( as I already mentioned , I just need to complete Linear algebra) What I was asking was guidance for mathematical topics beyond the standard sophomore curriculum

 

[Vanadium 50]

I have completed almost all of it

Bet you haven't. Bet you whipped right through it with little retention. We see this a lot here. People, mostly young people, try to go through the material 10x faster than everybody else and discover at the end they remember 10% as much as everybody else.

 

[crimeanwarcrimes]

Bet you haven't. Bet you whipped right through it with little retention. We see this a lot here. People, mostly young people, try to go through the material 10x faster than everybody else and discover at the end they remember 10% as much as everybody else.

Well I have been working at it for the last two years the only parts that I have left would be parts involving heavy use of linear algebra and I really don’t appreciate the condescending tone of yours I have been studying in an accelerated program in an Asian country which mostly covers a sognificant portion of undergraduate physics However I would still appreciate any valuable information regarding the matter

 

[Vanadium 50]

I really don’t appreciate the condescending tone

I'm sorry, I thought you wanted advice. What you really want is validation. OK, "you're 10x smarter than everybody else so you can go through the material 10x faster than everybody else". Is that better?

 

[S.G. Janssens]

However, I am still not proficient in various topics like Linear Algebra, Basic Analysis, and I quite often encounter various new things that I haven't even heard of.

The above implies that you are not ready for the mathematics typically encountered in graduate physics studies. If nevertheless you would like to bump your head with frustration, you could proceed with something like Reed's and Simon's Functional Analysis. (I would suggest to start with part 1, but who knows in your case.)

I have completed almost all of it and few things beyond that like Vector Calculus ( as I already mentioned , I just need to complete Linear algebra) What I was asking was guidance for mathematical topics beyond the standard sophomore curriculum

How can you complete vector calculus without proficiency in linear algebra and basic analysis?

Honestly, I don't understand why you are in such a rush. Create a solid undergraduate basis, then explore what kind of physics (specialization, mathematical, theoretical, experimental,...) you actually like most for graduate studies. Do you already have an idea?

 

[bhobba]

but you are looking to prepare for graduate physics courses?

There are advanced HS's that go into areas normally considered upper undergrad eg
https://www.basised.com/academics/curriculum/grades-8-12-curriculum/

Personally though I would take the graduate at grade 11 option, use any accumulated college credits, and complete my undergrad quickly in a combined undergrad/masters program which some schools offer - or maybe go to the UK where such is common.

Thanks
Bill

 

[bhobba]

My advice for what it is worth is two books. The first is Boaz:
https://www.amazon.com/gp/product/0471198269/?tag=pfamazon01-20

That is for the math used in physics. It can be tackled after US AP Calculus BC. But is not rigours.

For rigour my strong suggestion is from Matrix textbooks:
http://matrixeditions.com/5thUnifiedApproach.html

It too only requires AP Calculus BC - but beware - it is not an easy read - although very beautiful if you like math. It is used in a first year Honors course at Cornell.

I suggest starting with Boaz then the matrix textbook.

But I have to say while it is fine doing what you are doing, redoing it at whatever school you go to would still be beneficial. We are not all like Feynman or Fermi. Feynman basically started graduate courses second year, while Fermi was already doing work of PhD dissertation standard on entry to university. If you are at that level, professors are always on the lookout for such people, they will notice, and map out a path suitable for your advanced preparation.

Thanks
Bill

 

[bhobba]

How can you complete vector calculus without proficiency in linear algebra and basic analysis?

Probably from something like Strang:
https://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/

Although when I did it I had basic analysis, linear algebra and applied linear algebra under my belt.

Thanks
Bill

 

[jedishrfu]

Placing This thread into moderation until the original poster responds to PF warnings and contacts @Greg Bernhardt to change his name to something more acceptable.