Ready for first year rigourous calculus?

[emyt]

Hi, I'm going into my first year of university and I'd really like to know about how "ready" one should be before taking a first year rigourous calculus course (Analysis I). My current plan is to take random credits of courses that interest me and wait until next year before I begin taking the courses required for entry into the math specialist program.
To me, it's a foolproof plan - I could only become stronger in math if I had more time to self-study; but on the other hand, I'm very anxious to begin learning math from great professors.

At any time you could just scroll down and skip my life story:

Anyway, my main problem is that I never REALLY took any math courses in my high school years (I stopped at grade 11, and I didn't do so well). I was just a bad student and I didn't care about anything. But I took a year off after I graduated from high school, and during that year off (not the whole year though..) I became extremely interested in mathematics - I learned all that I could from books..et c. I also took some crash course in preparatory mathematics for the university level for about 2 months (we went through all of precalculus and went into calculus).
...
Anyhow..
... I was wondering what the "average" level of knowledge and ability a "usual" student would have before going into a first year undergraduate analysis course.

This is probably most of what I know:

All of pre-calculus (trig, inverse trig, logarithms..et c.)
Limits (epsilon-delta definition, calculating limits, "simpler" epsilon-delta proofs, proofs of the limit laws)
derivatives (proofs for differentiation rules, proof of derivatives like ln(x) = 1/x, calculating derivatives)
some random things like the division algorithm [ prove that if n is a perfect square, n divided by 4 will always a remainder of 0 or 1, prove that all q and r are unique to a set of a and b (where a = bq + r), prove that r < b)
Induction, strong induction and the well-ordering principle.

And right now I'm reading up more on algebra (groups and fields, vectors). I'm also reading from the Calculus book by Michael Spivak (haven't really gone too far yet, doing the exercises at the end of chapter one)

One of the things I'm most concerned about is that my basic "computational skills" are not as strong (very much so), I'm the kind of person who could overlook factoring a stupid polynomial because he thought that the +4 was a -4... I'm also worried about the "strength" of the problems I'm studying: maybe the solutions I've been coming up with are not as "clever" or maybe I haven't been trained enough in "word problems".. And finally my biggest concern is that my geometry skills are probably lacking immensely.

and I have another question:

Is it really necessary to take physics courses as a math student? The "course calendar" of my university even recommends that I at least take a first year physics course (and one of the more advanced levels). I'm glad that I've been lucky enough to catch up on a lot of things, but I think that there's no hope for me in physics. While I haven't really been introduced to physics, I feel like it could be something I'd be interested in. But I feel like I have no time and no direction - I wouldn't even know where to start..

I know it's a bit to read, but thanks for your time :)

 

[rubrix]

interesting...

lemme begin from bottom of your post.

Physics is literally an Applied Mathematics. If you are into Applied Mathematics, you most likely will like Physics, i know i do. Physics is not required for Maths major, not at all. But if you try it out, believe me it will open a new window of thinking/analyzing in your brain. And i would recommend you to do Calc based Physics rather than non-Calc based.

i wouldn't burn myself on low computational skills. Mine isn't that great. I know bunch Maths major who aren't good at that either, including Maths and Physics professors. Computational skills is least of your concern if you do pure Maths.

At my university we don't do undergrad analysis until our junior year (3rd year). Before that we do, Calc I (limit and derivatives), Calc II (integrals), Calc III (multivariable Calculus), and Introductory Advanced Mathematics Course (introduces functions, set theory, proofs like proof by contradiction & induction, module, congruence, equivalence, gcd etc). Those topics from introductory advanced mathematics courses come real handy in higher courses like Analysis and Abstract Algebra.

You do not need to know proofs for Calc I, Calc II, and Calc III. They are basically "plug and chuck" courses. Although they introduce formal definition of limit (epsilon definition) in Calc you most likely won't be tested on it. You won't need to prove derivatives rigorously, you don't do that until Analysis.
keeping that aside, what concerns me is that you are studying way ahead. I don't mean to discourage you from learning new stuff but looking into analysis w/o formal course in Calculus and studying abstract algebra (groups etc) w/o knowledge of module, congruence is off the track. If for some reason you are required to take Calc I or say PreCalc as your first course in college, you most likely will hate it as you already seem to know it. And I'm pretty sure they won't let you leap all the way to Analysis...there's a lot in between.

 

[snipez90]

Um, physics isn't really applied mathematics? Perhaps you meant it in the sense that mathematics is (obviously) important to understand if you want to do physics. Theoretical physics is more like pure math from what I've heard.

I agree good computation skills aren't required, but I would say that if you are not fluid with the basic computations in Spivak's first chapter (including the problems), you should invest more time into getting better at it.

I don't understand why "they" wouldn't let the OP into the analysis course. As you pointed out yourself, calc II and calc III don't really factor into how well you do in intro analysis. Besides, many universities do offer analysis to freshman and sophomores.

To the OP: you have a pretty good background for tackling Spivak's text. It is pretty much an intro analysis text (minus the basic topology). If you are referring to the kind of word problems that crop up in elementary algebra courses, there isn't really any of that. Don't worry too much about not being that good at geometry. You'll build up geometric intuition for the analysis concepts if you work through Spivak. Also, it's not necessary to do all of the problems in Spivak. A good effort on the ones that you find challenging or interesting is more useful than doing the ones you are fairly confident you could get if you worked it out.

 

[rubrix]

Besides, many universities do offer analysis to freshman and sophomores.

humm i didn't know that. I don't see how one could learn much in analysis w/o having done Calculus and at least one introductory course in advanced mathematics. Most likely the analysis course that is available is watered down to death.

Um, physics isn't really applied mathematics? Perhaps you meant it in the sense that mathematics is (obviously) important to understand if you want to do physics. Theoretical physics is more like pure math from what I've heard.

lower level courses like mechanics are i would say. There's velocity, acceleration that one does in calc I and there's an entire chapter in SHM which comes in DE course. Alike mathematics, Physics at higher level branches off to theoretical and applied (?). The latter one having much correlation to applied mathematics.

 

[emyt]

Thanks for the replies, the "Analysis" course is just calculus taught at a more thorough level.

Anyone have any opinions on my taking it next year plan? I just saw a "diagnosis test" and I'm not completely sure if I'm totally comfortable with that level (there was also a time limit to it).

 

[boboYO]

Hard to say, the difficulty of courses varies immensely between universities, but I would say that if you can work through spivak yourself and do some of the harder starred problems you should be fine. If you're having trouble with proofs, I've heard that How to Prove It A Structured Approach - Daniel J. Velleman is good for beginners.

And regarding physics, personally I find first year physics really boring and handwavy compared to maths but I do it anyway because I think it helps build mathematical intuition. It should be fairly easy for you if you can do proofs, I see a lot of people strugglign with physics 1 because they don't know how to use the given information to 'deduce' stuff.

 

[emyt]

what should I generally know before entering first year physics? I wouldn't be okay with just knowing math would I?

 

[rubrix]

you should be find. they'll teach you from basic.

 

[boboYO]

Yes, if it's a first year physics course they will start from the ground up. If you are familiar with vectors you should be fine.

 

[rubrix]

they even teach vectors. Usually chapter 1 of mechanics.

 

[emyt]

thanks, but I was looking through some physics textbooks and it looks like they use integration fairly early in the book? (I'm seeing it scattered around chapter 8.. et c in a 1056 page textbook)

What "level of education" would this class be: ? I could basically comprehend most of what he was saying (even though I haven't finished most of the video) without thinking too much, but it also looks like he assumed that the class knew a few things

 

[rubrix]

you'll see higher mathematics stuff in any physics course. You'll see partial derivatives (taught in Multivariate Calculus) as early as Physics I (mechanics) when prerequisites for Physics I is only Calc I (limits & derivatives). You'll also see Differential Equations in Physics I when you talk about Simple Harmonic Motions. It's good to know but don't torture yourself over it. I know Physics professors say something alone "this is partial derivative and this is ODE...you can learn about them in higher maths course but for now let's take it (w/e is the consequence of using them) as a fact. You won't be required to know how to use partials and ODE."

 

[emyt]

Your professor might simply ignore it, tell you it's a given fact. In either case you are not expected to know those stuff in detail.

aww, that won't be very fun

 

[rubrix]

fact is not everyone doing Mechanics (and Electricity and Magnetism as well) is interested in Physics. Most are doing it to fulfill prerequisites for other majors like engineering, premed, bio, chem w/e. I know in our Physics I class there was ONLY ONE Physics major...who later from what i hear ended up switching to Mathematics or something else.