Which math subjects are good for self-study and how should I go about it?

[megatyler30]

Hi, I'm a Junior in high school. I just finished Calculus 3 (and got an A) last semester and I'm taking Differential Equations next semester. Also (if I get into this free math program), I will be taking Linear Algebra and Number Theory with Cryptology this summer. Anyways my question is what math subject would be good to self-study (that I wouldn't be repeating much by taking the classes I mentioned)? Would Discrete Mathematics be a good idea? Also how would you recommend I self-study, just textbook and problems? Also please suggest good textbooks and other resources. Thank you

 

[R136a1]

If you already know calculus, then you have a ton of options. Since you already done calc III and you're a junion, I'll assume you want books that are pretty challenging.

Here are some things you could do:

Linear algebra (but you already will do that, judging from your posts)
Abstract Algebra: Artin's book is excellent https://www.amazon.com/dp/0132413779/?tag=pfamazon01-20

Differential Geometry: https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20
Logic: https://www.amazon.com/dp/0387942580/?tag=pfamazon01-20
Set Theory: https://www.amazon.com/dp/0824779150/?tag=pfamazon01-20
Analysis: perhaps working through Spivak's book would be a good start https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20 or berberian: https://www.amazon.com/dp/0387942173/?tag=pfamazon01-20

All of these books are possible for you to do with a knowledge of basic calculus. But that is assuming you are a bit comfortable with proofs. If not, a basic proof book such as https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20 should do.

 

[thelema418]

I don't know what Calculus 3 includes as content, but I presume that it is computation based (which is typical in terms of instruction) You may want to do real analysis. This repeats material you will have seen in all your calculus courses, but it is a deeper investigation into the concepts and proofs. It will introduce a lot more with set theory than you have probably seen in Calculus.

If you want to continue what you have done in calculus, you can probably move into Complex Variables. Doing Calculus in the complex plane is extremely useful.

For differential equations, transformational geometry is very helpful, especially if you do partial differential equations. You can explore this with software like Geogebra.

Statistics is also a good thing to look into. I'm not certain what background you have with that, but there is so much to learn in that field. Also, learning SPSS or R is useful.

My last recommendation is that you choose something that you think will be useful to you. If you want to do biological research, then you might want to examine a text on denography. If you are interested in economics, you might want to look at applied math for finances. etc.

 

[megatyler30]

Thank you a lot both of you and by the way Calculus 3 isn't computational calculus but rather multivariable calculus. And the intended majors are chemical engineering and chemistry.

 

[lavinia]

At some point you must learn Complex Analysis. Sooner doesn't hurt.

 

[megatyler30]

Don't I need to have differential equations down first?

 

[lavinia]

Don't I need to have differential equations down first?

Not for Complex Analysis. You will learn about solutions of the two dimensional Laplace equation as you go along.

BTW: I found it useful to learn some theory of electricity and magnetism to help with Complex Analysis.

 

[megatyler30]

Okay thank you, I'll check it out when I can, but won't I still need to do real analysis first ?

 

[megatyler30]

I ended up getting into the summer math program so I will be starting Linear Algebra and Number Theory with Cryptology soon. Finally I'm done with all my Junior year testing. Currently I'm trying to decide to do either Abstract Algebra or Real Analysis now that I finally have some time. The part of my initial question about how to go about independent study was not really answered and an answer to that would be helpful. Especially how to go about problems when I get stuck and where to get verification of answers if it is not in the back of the book.

 

[micromass]

Especially how to go about problems when I get stuck and where to get verification of answers if it is not in the back of the book.

Post the problems on this forum and people will help you.

 

[verty]

What about probability and combinatorics? Game theory? Some of the applied math might be quite interesting.

Abstract algebra or real analysis, I would do both in parallel. For real analysis, you can read the free book by Trench if you like. But whatever you do choose, avoid Rudin. Amazingly his second book seems better than the first, and it too is unforgivably bad.

If you use Artin for algebra, there are Harvard lectures available by Benedict Gross, search for math e222 videos. It may help to use them because there is a lot to learn.

 

[TitoSmooth]

Get a better grasp on euclidean geometry. Ie kisselev planimetry volume 1 and sterometry volume 2 for solid geometry.

 

[x BlueRobot]

I would study what your interested in, but I would suggest studying something which requires proof writing and some thought rather than just completing equations, since you'll eventually will need to start writing proofs at some point.

 

[thelema418]

I ended up getting into the summer math program so I will be starting Linear Algebra and Number Theory with Cryptology soon. Finally I'm done with all my Junior year testing. Currently I'm trying to decide to do either Abstract Algebra or Real Analysis now that I finally have some time. The part of my initial question about how to go about independent study was not really answered and an answer to that would be helpful. Especially how to go about problems when I get stuck and where to get verification of answers if it is not in the back of the book.

Congratulations on your acceptance! Abstract algebra and real analysis will probably entail a lot of proof-based questions.

Proof will be challenging. Proof is a justification that has been examined by a community of peers for its reasonableness. At the same time, a proof must satisfy self-doubt. In terms of self-study, I think it is difficult to do proof-based activities fully.

I think it is beneficial to work with your peers. If distance is a problem, create a Google site so that you can productively work on problems with others. If you are stuck, post problems and your solution attempts on the homework board here.

Also, textbooks notoriously use "obviously" or "it is obvious that" in a proof. It is a good strategy to question segments of a proof that use that phrase. Additionally, don't assume textbook proofs are the best and only proof.