What should I study in my free time?

[GuitarStrings]

Okay, so my vacations are going to be coming up soon, and I know I am going to eventually get bored and want to start doing math. (None of that enjoy your life stuff please, that goes with saying. Plus, Maths is fun )

What I've essentially done till now is high school math.

Syllabus:

Algebra (Series, counting principles, mathematical induction, complex numbers, )
Functions and equations (Basics)
Circular functions and trigonometry (Identities, essentially)
Matrices (Addition, subtraction, multiplication, determinant determination, applications to simultaneous equations)
Vectors (Cross and dot product, vector equation of a line and plane; distinguishing between coincident, parallel, intersecting, and skew lines;)
Statistics and probability (Various distributions, permutations and combinations, conditional probability, etc)
Calculus [Single Variable] (Limits (Informally), basic derivatives, basic integration (parts and substitution))
Series and Differential equations (First order separable differential equations, proving sequences as convergent or divergent, basic improper integrals, power series, taylor and McLauring series, l'Hospital rule, slope fields)

I was thinking of doing limits formally, and learning tensors and continuing with more difficult differential equations.

But tensors may be a bit ambitious. Any other recommendations?

(P.S. If anyone wants a more detailed coverage of my syllabus, (probably not, eh? ) http://www.google.co.in/url?sa=t&so...g=AFQjCNFgT-gWukHzjSGzf7FLJkSi-zRm3A&cad=rja"it is)

 

[wisvuze]

I'd say to go with some linear algebra, it may be your first look at some more "abstract" mathematics. Besides, you may not really have an appreciation for tensors if you don't look at linear algebra first

 

[GuitarStrings]

From what I understand I can pick up the linear algebra WHILE doing the tensors? Unless I'm horribly mistaken?

 

[micromass]

From what I understand I can pick up the linear algebra WHILE doing the tensors? Unless I'm horribly mistaken?

If you're talking about the tensors that I'm knowing, then I would recommend doing the linear algebra first, and then doing the tensors. Doing them together, may yield confusion.

Some other ideas: discrete mathematics, set theory,...

It actually all depends on what you want to do with the mathematics. Do you want to go into physics, mathematics, engineering?

 

[GuitarStrings]

Engineering. But I don't really care if what I study relates to the field.

And for what its worth, I enjoyed calculus the most throughout school, although that may be because it was the only close to challenging thing we did?

I like things where you need a creative step which makes everything fall in place, if you get what I'm saying? Also, I enjoy spending a lot of time trying to make sense of a concept because when it finally hits me, its a really amazing feeling.

Set theory, not my thing. I looked into discrete mathematics superficially, but as of now, linear algebra seems more of my thing.

 

[micromass]

Yes, if you want to go into engineering, then set theory and discrete mathematics are pretty useless.

So I think you can't go wrong with linear algebra. Or maybe do some multivariate calculus (partial derivatives, multiple integrals and the like). All these topics will be very benificial to an engineering studies! And I think you'll enjoy them...

 

[GuitarStrings]

Thanks a lot. Are there any specific books you recommend?

 

[micromass]

I think the standard references are Apostol for calculus, and Shilov for linear algebra. But these texts are for mathematicians, so I doubt you will find many applications in these texts.

Other (free) texts can be found on the following site:
Linear algebra: http://hbpms.blogspot.com/2008/05/stage-2-linear-algebra.html
Calculus: http://hbpms.blogspot.com/2008/05/stage-2-calculus.html

But again, there probably are texts which are more application-focused, and thus better for you...

 

[deluks917]

I suggest Loomis and Sternberg "Advanced Calculus." It has a good mix of theory and application and is a good place to learn some linear algebra. It is also considered very diffcult to read but you seem smart and like you want something both challenging and rewarding. This book was historically used for Math 55 at harvard and has a wonderful chapter on Classical Mechanics.

 

[flemmyd]

Don't set theory/discrete math help a lot for computer engr majors? The seem like they could also be useful for an EE major/focus on signal processing...

I'm tempted to say linear algebra.
In terms of books, just try math methods by Boas. Will have a section on almost anything that interests you and is geared towards non-math majors who just need the math tools to do their physics/engr.

 

[mathwonk]

remember music, literature, fine food and art. you will still need to woo women, and decorate your apartment.

 

[barcelona5]

honestly, I'd tell you to skip the math and just go for some sort of accounting of finance class - that will actually be useful in the real world!

 

[snipez90]

Dude I highly suspect you're trolling.

Anyways do linear algebra and multivariable calculus at the same time. It turns out they're pretty related. Stewart's Calculus text is fine for multivariable calculus. Rigorous multivariable calc theory is either pretty lame (those parts that just extend basic one-variable theorems) or a dinosaur (differential forms). Provided you're in Euclidean space, the nature of multivariable calculus is computational, and this will be evident if you take complex analysis or a partial differential equations course later on.

It is also a pretty good idea to expose yourself to basic ODE theory. You can start learning general first order equations. I recommend sosmath.com for this. You can save second order (and by simple extension n-th order) equations for after you know a bit of linear algebra.

 

[aspiring_one]

multivariable calc. and linear algebra and then maybe DiffEq

have you taken a geometry class?

 

[osnarf]

Set theory is not useless at all if you intend on doing any rigorous treatment of... anything? Discrete mathematics can be useful in serious programming, and you will need to be able to program at least on some level.

 

[eXorikos]

remember music, literature, fine food and art. you will still need to woo women, and decorate your apartment.

I see your point. Recently I've been going more for economics in my spare time than physics or maths. Although I don't need to woo women anymore. ;)