# Using Spivak and Apostol Calculus in parallel?

1. Aug 21, 2014

### EmSeeSquared

Hi all,

So I got 2 months exactly before my exams and since I want to start from 0 again because I realized my base is not very well founded. So I'll take this weekend to go through my pre-calculus again and start my Spivak-Apostol schedule Monday. I need to polish my maths, start with the basics again.

I use Stewart in my college and I don't really like it. It has tons of exercises, that's good but Spivak and Apostol have loads of proofs and are more rigorous so I will be studying these 2 books only and in depth.

I'm making a plan to finish both Spivak and Apostol in 1.5-2 months. I'm not sure if it's doable(at 5 hours/day). I'm also planning to cover the exercises in Stewart.

However I'm not sure how to go about reading both of the at the same time, since Apostol has a different style altogether. Any help is appreciated.

Also, I would like to know how to go about in solving multiple choice questions. I always get stuck on the true-false questions and the problem is we have one of the final papers which is just multiple choice. I'm guessing I need to know all my concepts at my finger-tips and that's my main problem. I need to polish my concepts and memorize them. What would be the best way around the concepts? I mean, how to I go about knowing what to know?

2. Aug 21, 2014

### verty

My advice is to read through each, summarising the text and doing no problems. At the end, try to tie it all together, see what fits where. Then hit the problems, go through them nice and quickly, skipping any that are hopelessly too difficult.

3. Aug 21, 2014

### EmSeeSquared

That's a good idea I thought about, but won't I be missing on practice? On the bright side, if I manage to do this, it means I'll be really good with my concepts, since I'll be using them after weeks to do the practice problems.

How about learn the concepts do 3-4 simple questions to get the topic and move on to reading and summarizing?

Oh about the latter, when you say reading and summarizing, does it mean taking simple short notes? I never did this before, so I'm not sure how to go about this. I would appreciate if you could give some tips here :)

Thanks

4. Aug 21, 2014

### Acid92

Spivak and Apostol are generally regarded to be around the same level in difficulty and rigour but most people seem to prefer Spivak for its style (Ive only ever used Spivak so I cant comment on Apostol but Spivak was definitely among the best texts Ive ever done). Trust me, you won't be able to complete both in 2 months (unless youre one of those prodigies) if you intend on completing most of the exercises (and that is the most important thing really). Instead of redundantly using Apostol with Spivak, I would recommend using "Numbers and Proofs" by Reg Allenby to complement Spivak. This gem is a learn-how-to-do proofs text contextualized in very elementary number theory that Imperial College London prescribes to its prospective Math undergrads and unlike most of those proofs-books that are basically just basic set-theory and formal logic texts, this will really get you doing proofs.

5. Aug 22, 2014

### verty

To your first question, you will still be doing problems, just not immediately. To your second question, I don't like that idea, I mean doing 3-4 simple questions, that is going to give superficial confidence, it'll seem like you know it when you really don't. But summarizing only, you would have no confidence that you really understood anything. But confidence is bad, you want to have no confidence until you understand it very well.

6. Aug 22, 2014

### EmSeeSquared

Let's say I do only Spivak's book, will it give me a good background to proceed to R.A, LA?

7. Aug 24, 2014

### mathwonk

this cannot be done probably in 2 months. i suggest just learn what you can rather than setting unreasonable goals. the point is to learn something, not to read so many pages.

8. Aug 24, 2014

### verty

Learning is a process of placing the knowledge in your brain in a very connected state, where everything fits together and is in its place. If you are someone like Mathwonk who has a very good memory, it is easier, you can learn in a serial way and things will get processed in the right way. But if you are someone like me who doesn't have a good memory, it'll be more difficult to learn in a serial way.

And you won't have a teacher to show you where things fit, so I think it is more helpful to read ahead without doing the problems, although you can look at the problems and think how you might do them. But the point is not to build up too much, too soon. I suppose it's like building a puzzle, one always builds the outline first. Building a puzzle without building the outline first would be much more difficult.

9. Aug 24, 2014

### EmSeeSquared

I've taken a look at the books and since Apostol seems to cover more than Spivak, I'll be working only though Apostol. Since I use Stewart in college. I'll just do the exercises in Stewart as a complement to Apostol. As for Spivak, I'll just read it "like a novel"(not doing exercises). Apostol seems to be better with the theorems, proofs and it seems better when I read it. But I guess I should brush up a little bit on my pre-calc before tackling it. I guess I'll take 2-3 days to brush on the pre-calc and start Apostol.

Question, is whether it will be enough for me to go for my exams in exactly 2 months. I have some calculus knowledge, but it's from Stewart(which I don't really appreciate except for it's tons of exercises). Basically I'm trying to get my concepts and proof writing right and Apostol seems to do a better job than Spivak on that one. Stewart didn't give me a good foundation to be honest. I came to college without knowledge of proof writing(except simple trig proofs). So it was hard for me in the beginning. What saved me in the exams so far was just because I could remember the proofs and write them(they ask us to memorize some proofs) and because I did the questions in Stewart. But I still feel my math isn't good.

What do you guys think?

10. Aug 24, 2014

### verty

You are doing a proof course? I would choose Spivak in that case, it really confronts you with the need to prove things and it starts out with easy proofs in chapter 1. Some of the questions are scary, I would just skip them.

Best of luck to you.

11. Aug 24, 2014

### EmSeeSquared

It's not really a proof course but we do have some proofs. Induction, contradiction and the rest are those in the chapters, such as MVT, FTC, some proofs of derivatives(first principles) and some conbinatorics proofs(very simple and easy). And maybe some more. But most questions are solve, and conceptual questions(this is my worst part). I really struggle with conceptual questions at times and this is what I want to get good at. Based on this is Spivak still the way to go?

P.S Normally we are asked to just remember these proofs and we get 1 question where we are given a choice to write one of them. Basically it's just remembering the proofs which takes 5 minutes per proof. However, I want to be able to apply these proofs.

12. Aug 24, 2014

### verty

I think these are not the right books for you (or perhaps Apostol is, I don't know), try this instead. If you finish with that and still want more, bump this thread or start a new one. But that will be something extra.

Last edited: Aug 24, 2014
13. Aug 24, 2014

### EmSeeSquared

The famous MIT lectures. I have the whole series on iTunesU. I started watching some but stopped half way I guess, since I thought it was taking too much of my time. I guess I'll watch the rest now :)