Analysis Learning: Spivak or Stewart? Self-Taught Student

[woundedtiger4]

I have realized that almost everyone on this forum talks about the calculus by spivak, and from the arguments in it's favour make it unbeatable, it seems that nobody can learn analysis without reading that book. I am studying ONLINE at Paul's notes and Calculus by James Stewart, is my approach/route towards analysis is weak? do I have to restart calculus by reading Spivak?
P.S. I am a self-taught student, and I have nearly finished calculus 2.

 

[micromass]

Depends on what you want to do later. If you want to do physics, then you don't necessarily need Spivak.
If you're into math, then you'll need to study a rigorous calc/analysis book sooner or later. And Spivak is a very good book.

 

[woundedtiger4]

Depends on what you want to do later. If you want to do physics, then you don't necessarily need Spivak.
If you're into math, then you'll need to study a rigorous calc/analysis book sooner or later. And Spivak is a very good book.

I want to do the pure maths such as real analysis particularly measure theory (in context of financial maths).
Isn't good if later I study some soft book on analysis and then the hard book on analysis, instead of restarting again the calculus?

Thanks

 

[micromass]

I want to do the pure maths such as real analysis particularly measure theory (in context of financial maths).
Isn't good if later I study some soft book on analysis and then the hard book on analysis, instead of restarting again the calculus?

Thanks

Analysis IS restarting the calculus, but then rigorously. At least, that's how it begins.

You don't need to read Spivak, you could also read an actual analysis book. But analysis books are often somewhat harder.

 

[QuarkCharmer]

I learned Calculus from Stewart, Pauls Online Notes, OCW.MIT.edu, and Khanacademy. I have a copy of spivak's calculus. I worked through 2 chapters and did ALL the problems. It took me months to get that done.

If spivak was such a fantastic book for learning calculus with no previous knowledge, why is it used in courses on analysis? I wouldn't sweat it at all. If you can read through Spivak, you will likely find it interesting. The problems are REALLY challening, exponentionally so, in relation to the sets found in Stewart, but I wouldn't sweat your understanding of calculus based on that fact, or the in-depth-ness of Spivak. You'll meet those in-depth topics eventually anyway in other courses most likely.

Besides, when you do get around to reading Spivak, or an analysis book, you will at least have a pretty good idea what the concept is doing when you work through the analysis/proofy stuff.

 

[Robert1986]

IMHO, there is something of a spectrum from "pure calculus" (I think I just coined a term) through "pure analysis." On the left side (the "pure calculus" side) you have books with titles like Calculus For XXX where XXX might be something like "Dummies" or "Business Students" (but I repeat myself, jkjk). A little to the right of that (toward the "pure analysis") are books like Calculus For Engineers. Books in these two categories are motivated almost 100% by "real-world" problems and give very few (if any) good or detailed proofs. Essentially everything is "beyond the scope of this book."

Stewart falls somewhere just slightly to the left of the middle of the spectrum. Nearly everything is motivated by physical problems, but he gives a satisfactory treatment of the important topics where possible. In addition, the appendices are very helpful and (I believe) include a proof of FTC.

Spivak is on the right side of the dividing line. Nothing (that I remember) is motivated by anything other than math. This is a really good book, but it would be a tragedy to give this book to a first-year calc student (unless the student is exceptionally bright.) And I believe Spivak says something to this effect in the preface. At the very least, one who reads Spivak needs to be familiar with the mechanics of limits/derivatives/integrals even if he doesn't understand them completely (and no one who has only had calc understands them completely.) A little more to the right are books like Elements of Real Analysis by Bartle (this is the one we use at my school) and "Baby Rudin." These books are much more than a book on Advanced Calculus but much less (especially the Bartle Book) than a "Pure Analysis" book. Bartle doesn't go into abstract integration but instead confines himself to R^n. IMO, this type of book is perfect for an undergrad-level analysis course.

On the "pure analysis" side is "Big Rudin" (Real and Complex Analysis.) This is a good book, but I really wouldn't recommend it for anyone who hasn't read at lest one of the books from the previous section. So, since you have essentially completed Calc. II, I would recommend that you "complete" calc III (the multi-variable stuff.) While you are doing that, it would make sense to begin reading Spivak and working on the exercises. Then, pick up Baby Rudin or the Bartle book (or whatever your school uses for the Analysis course.) Also, you say you want to go into "pure maths" particularly in the context of "financial maths" I don't really know how that is possible, but if you want to go into pure maths, you need to start reading Algebra books. The one we used at my school is "Topics In Algebra" by Hernstein. This is a good introductory book to the subject. The "Big Book" in this field seems to be Abstract Algebra by Dummit and Foote. This is what my school uses for the Grad-Level Algebra sequence. It is good, but it is rigorous and gets pretty difficult at some places. I wouldn't recommend this as a first book on Abstract Algebra.

 

[woundedtiger4]

Thank you so much for very detailed answer of my question.
Actually, I want to study the topics of measure theory & probability theory such as "Probability spaces and probability measures. Random variables. Expectation and integration. Convergence of random variables. Conditional expectation. The Radon-Nikodym Theorem. Martingales. Stochastic processes. Brownian motion. The Itô integral." The purpose to study is to understand the formal basis of abstract probability theory, and the justification for basic results in the theory, and to explore those aspects of the theory most used in advanced analytical models in economics and finance. And then I want to move towards Stochastic Analysis in which I can study martingales (in discrete and continuous time), including limit theorems, and Brownian motion, Stochastic integration (a brief introduction to Itô calculus including Itô's lemma theorem and stochastic differential equations).

Currently, I am studying Calc3 (just started about an hour ago) also a book by Stirling "Mathematical Analysis and Proof" (which is for 1st and 2nd year's BSc (UK) students), but after reading couple of chapters everything started going over my head therefore now I have ordered a new book by Mary Hart

http://books.google.co.uk/books/about/Guide2_analysis.html?id=D3N4QgAACAAJ&redir_esc=y

the review says that it's very good after A-level maths (class 12)

Can you please check Paul's Notes on calc1 & 2 and give me your opinion on it.

I am surely going to do the same thing like you said, while studying calc3 I will study Spivak as well from the beginning as by doing so I will not only able to understand analysis but also I will also do the revision of what I have already studied from other sources.

Once again many thanks for sharing your thoughts and I hope to hear soon from you.

 

[woundedtiger4]

Actually, I am not very good in studying two things simultaneously therefore will it not be good if I start Spivak and once I have finished it then I start calc3? As then I will also able to study linear algebra along calc3.

 

[Robert1986]

Actually, I am not very good in studying two things simultaneously therefore will it not be good if I start Spivak and once I have finished it then I start calc3? As then I will also able to study linear algebra along calc3.

I LOVE Paul's Online Math Notes, as well as Khan Academy (I believe you already mentioned this.) IMO, one can get a pretty good Calc I-III/ODE education from these two sources (including, of course, working exercises.) Throw in the MIT OCW stuff as well. In fact, if one were to just study these three things he would have a math education that is roughly equal to the engineers where I go to school (Georgia Tech.)

As far as not being good at studying two things at once - and I don't know how to put this lightly - get over it. At some point you are going to HAVE to study more than one thing at a time. And when you get done with school you are going to be doing more than one thing at a time. So, you are just really going to have to make disciplined plans and stick to them. Make a "study date" with yourself, if you have to. Say something like "At 2pm I will go to the library to study Spivak for 3 hours." Leave your phone at home, leave your computer at home. Just go and sit with a copy of Spivak (and perhaps a thermos of coffee.) Then, do something similar with Calc III.


If you have truly been able to self teach yourself through calc II, you probably have the talent to do what I suggest. Also, you need to be doing Linear Algebra NOW. It is quite different than any math you have done up to now, and it will take a little while to really feel comfortable with it. Also, having a knowledge of Linear Algebra will help you in Calc. III which, in turn, will help you understand Lin. Alg. a little better. Furthermore, if you want to be a quant (and it sounds like you do) you have no choice: you MUST take PDEs, which means you must take ODEs. Since you really need Lin Alg for ODEs, you REALLY need to start Lin Alg now. I transferred into my current college already haven taken Linear Algebra and Calc II, but at my college, Calc II and Linear Algebra are combined into the same class.

So, bottom line is this: start studying Spivak and Calc III and Linear Algebra now. Give yourself a deadline (are you currently in school anywhere?) of when you should be done. Then move on to Calc III and ODEs.

 

[woundedtiger4]

.

Once again many thanks for your prompt and definitely a very comprehensive reply.

Actually, I have done an undergrad in computer science but that was like more than a decade ago and in the practical life (industry) I was just a coder never used maths or algorithms just followed the given instructions :( and therefore I almost forgot what I learned in my BSc and plus I am now fed up of what I have been doing since so many years. Therefore, I have planned to study at MSc level (postgrad) & I am more interested in a quantitative subject so that I can prepare myself as a quant in IB, I have already secured an admission at the 3rd top ranked institute in UK perhaps my BSc grades were good (although I didn't study at top university) & I had good references as well or my gmat score was good but anyhow now I have unconditional offer letter (sep2012) which I received in January, and then I left my job and started studying on my own. I had never studied calculus in my life but somehow I managed to finish calc1 in one month and calc2 in 15 days (both on Paul's Notes including the complete practice set both given by Paul and Stewart), I usually wake up around 12 pm (afternoon) and then in an hour or so I start studying till 5 or 6 pm, and then I take break and then restart studying at 9 or 10pm and finish roughly around 4 or 5 pm, that's my routine. I don't have much of guidance all I have been told by my prospective university that I should be very good in pure mathematics in order to take "pure mathematics" therefore I checked several universities websites for prerequisites and found that the things work in math in the following order:
Stochastic Analysis<-Probability & Measure theory / Stochastic process <- Real Analysis <- Mathematical Analysis <- Analysis <- Diff Eq <-Linear alg <- calc3 <calc2<calc1. So now I have four more subjects to study before September 2012 so that then I can manage to study the subjects I want to study otherwise I will not only understand nothing in the class but will also fail the whole course. I don't have any guidance at the moment in terms of teachings therefore I am using internet as a guru and listening the valuable thoughts of several people such as yours. Perhaps, mostly people after reading this will make fun of me because I am going to study a subject that requires a certain level of mathematical maturity and loads of experience (in terms of by spending more & more time on zillions of problems) but I am inspired by Ramanujan who had no formal education in maths but yet he made huge contribution in pure maths, and if he can do this almost a century ago when there were very limited resources of information then why not me when I have loads of information sources such as internet etc. Plus, I have a will power, and I am very consistent towards my goal.
I will really like to ask you one thing that what is exactly mathematical analysis (in human language) I mean that I have almost finished the 15 pages long topic on functions in Spivak's book, I just learned what I have already learned by studying paul, stewart, and loads of other sources. Spivak has just used more words and less mathematical notations.
You are quite right that sooner or later I will have to adapt the habit to study more than one subject, I will surely make a new plan and will try to follow the divide & conquer policy.
I will appreciate your further guidance.

 

[micromass]

Impressive! But do you make enough exercises??I hope you're not just reading the theory while doing nothing yourself. You really need to make LOTS of exercises on the material.

Furthermore, I have some comment on your study plan. You want to study probability theory after studying real analysis. This means that you will study the formal and rigorous version of probability. I want to warn you that this is insanely difficult without previous knowledge of probability. I recommend you reading a probability book first that is introductory. I recommend the wonderful book "understanding probability by Henk Tijms.

Studying measure theory is good (and highly recommended). But I would study it in the context of probability theory. The book "probability and measure" by Billingsley is truly THE bible of rigorous probability theory. HOWEVER, you will need a very firm grasp of basic probability theory (a book like Tijms should be enough) and of real analysis (you should be acquainted with at least Spivak and a bit topology/metric spaces).

Don't waste too much time on Calc III. A lot of things are useful, but a lot of things aren't. You should now things like partial derivative really well and multiple integrals. Things like vector calculus (integration on surfaces, Stokes theorem, line integrals, etc.) can safely be dropped if you don't have the time.

The hardest thing for you will be to get acquainted with proof-based mathematics. I understand you're reading non-proofy calc books now. This is good, but you will find the transition to the proofy real analysis books to be quite a shock. This is exactly why you need to read Spivak. You already know all the material in Spivak, just not how the material is presented. That way you can focus on proofs and rigor all the way.
Furthermore, if you completed Spivak, then you don't really need much real analysis anymore. The only thing you should know after completing Spivak would be things about topology and metric spaces. Reading a real analysis book would still be a good idea though.

 

[Robert1986]

Once again many thanks for your prompt and definitely a very comprehensive reply.

Actually, I have done an undergrad in computer science but that was like more than a decade ago and in the practical life (industry) I was just a coder never used maths or algorithms just followed the given instructions :( and therefore I almost forgot what I learned in my BSc and plus I am now fed up of what I have been doing since so many years. Therefore, I have planned to study at MSc level (postgrad) & I am more interested in a quantitative subject so that I can prepare myself as a quant in IB, I have already secured an admission at the 3rd top ranked institute in UK perhaps my BSc grades were good (although I didn't study at top university) & I had good references as well or my gmat score was good but anyhow now I have unconditional offer letter (sep2012) which I received in January, and then I left my job and started studying on my own. I had never studied calculus in my life but somehow I managed to finish calc1 in one month and calc2 in 15 days (both on Paul's Notes including the complete practice set both given by Paul and Stewart), I usually wake up around 12 pm (afternoon) and then in an hour or so I start studying till 5 or 6 pm, and then I take break and then restart studying at 9 or 10pm and finish roughly around 4 or 5 pm, that's my routine. I don't have much of guidance all I have been told by my prospective university that I should be very good in pure mathematics in order to take "pure mathematics" therefore I checked several universities websites for prerequisites and found that the things work in math in the following order:
Stochastic Analysis<-Probability & Measure theory / Stochastic process <- Real Analysis <- Mathematical Analysis <- Analysis <- Diff Eq <-Linear alg <- calc3 <calc2<calc1. So now I have four more subjects to study before September 2012 so that then I can manage to study the subjects I want to study otherwise I will not only understand nothing in the class but will also fail the whole course. I don't have any guidance at the moment in terms of teachings therefore I am using internet as a guru and listening the valuable thoughts of several people such as yours. Perhaps, mostly people after reading this will make fun of me because I am going to study a subject that requires a certain level of mathematical maturity and loads of experience (in terms of by spending more & more time on zillions of problems) but I am inspired by Ramanujan who had no formal education in maths but yet he made huge contribution in pure maths, and if he can do this almost a century ago when there were very limited resources of information then why not me when I have loads of information sources such as internet etc. Plus, I have a will power, and I am very consistent towards my goal.
I will really like to ask you one thing that what is exactly mathematical analysis (in human language) I mean that I have almost finished the 15 pages long topic on functions in Spivak's book, I just learned what I have already learned by studying paul, stewart, and loads of other sources. Spivak has just used more words and less mathematical notations.
You are quite right that sooner or later I will have to adapt the habit to study more than one subject, I will surely make a new plan and will try to follow the divide & conquer policy.
I will appreciate your further guidance.

Well, I wouldn't recommend going exactly in this order. If it took you 15 days to get through Calc II, you can pretty much take a weekend on CalcIII. I think probably the main thing you need to get out of that is partial derivatives and multiple integrals (since you MUST take PDEs since stuff in finance has many independent variables.) Now, doing probability after all that other stuff makes me think that this is going to be a REALLY theoretical class. It is kind of like reading a real analysis book first instead of a calc book first. Read a basic book on probability. We used Probability and Statistical Inference by Hogg and Tannis in my Intro to Stat class; this is the book the Society of Actuaries recommends for the first actuarial exam, btw. I also used a book: Mathematical Statistics by Hogg in my Intro to Math. Stats class. In fact, a book on Mathematical Stats is probably a pretty good book to start with; just keep an intro book handy. But definitely get a Prob book that is calc-based. I know you don't have stats listed up there, but it is going to be a very important topic to know as a quant.


As for Mathematical Analysis, I must confess that I really don't know. I mean, you have Analysis, Mathematical Analysis and Real Analysis all listed. I don't really know what the difference between them is; I always figured they were synonyms. Anyway, it is just a really theoretical treatment of calculus (well, some people might have a problem with that description, but it is good enough).

 

[woundedtiger4]

Impressive! But do you make enough exercises??I hope you're not just reading the theory while doing nothing yourself. You really need to make LOTS of exercises on the material.

No, no, I am also doing practice of it for example I have done all the problems from Paul's extra in clac1 and Stewart's clac1&2 (roughly more than 2 thousand questions from limit to vectors) arghhhh I hated vector part, and I really struggled to understand the concept of that but someone on openstudy told me to study that too therefore I finished it :(
Yeah, I have clearly seen the difference of using abstract methods in Spivak. BTW, do I need to study abstract algebra, or linear algebra as well as someone told me that vectors + linear algebra is really important in stochastic.

What exactly I should study in order to manage these courses, somehow I managed to get the current year's lectures notes on these two subjects, I will really appreciate if you have time then take a look at them and please give me your suggestions about my reading list (pm me your email address).

Thanks

 

[micromass]

No, no, I am also doing practice of it for example I have done all the problems from Paul's extra in clac1 and Stewart's clac1&2 (roughly more than 2 thousand questions from limit to vectors) arghhhh I hated vector part, and I really struggled to understand the concept of that but someone on openstudy told me to study that too therefore I finished it :(
Yeah, I have clearly seen the difference of using abstract methods in Spivak. BTW, do I need to study abstract algebra, or linear algebra as well as someone told me that vectors + linear algebra is really important in stochastic.

Here is the links of the courses that I want to study in future:

http://www.lse.ac.uk/resources/calendar/courseGuides/MA/2011_MA411.htm

http://www.lse.ac.uk/resources/calendar/courseGuides/MA/2011_MA414.htm

What exactly I should study in order to manage these courses, somehow I managed to get the current year's lectures notes on these two subjects, I will really appreciate if you have time then take a look at them and please give me your suggestions about my reading list (pm me your email address).

Thanks

You don't need abstract algebra at all, so you can safely drop that. You will need linear algebra though. Furthermore, linear algebra offers you a chance to be acquainted with proofs.

The links you provided are standard upper-level probability courses. I recommend that you study the following things:

Calculus (level Spivak should be good + a little bit of multivariable)
Linear algebra (I like the book by Friedman. Check my blog: https://www.physicsforums.com/blog.php?b=3206 )
ODE's and PDE's (Boyce and Diprima is a good first introduction, although it covers only ODE's)
Basic probability theory (Henk Tijms is good, as I mentioned)
Real analysis (The book by Pugh is very good)

This are necessary prereqs (although ODE's and PDE's won't be necessary for the first course, but you'll need them eventually). Study them well.

If you have time, you can study measure theory. The book "Lebesgue integration on Euclidean space" is wonderful).

Feel free to PM me any time if you have any questions.

 

[woundedtiger4]

You don't need abstract algebra at all, so you can safely drop that. You will need linear algebra though. Furthermore, linear algebra offers you a chance to be acquainted with proofs.

GREAT.

You have provided me a wonderful reading list. I have started understanding Spivak but will you recommend that at same time while reading Spivak if I study your recommended linear algebra and Paul's calc3 (actually I understand his notes easily).

 

[micromass]

GREAT.

You have provided me a wonderful reading list. I have started understanding Spivak but will you recommend that at same time while reading Spivak if I study your recommended linear algebra and Paul's calc3 (actually I understand his notes easily).

Yeah, studying it simultaneously should be ok. In fact, I always liked to study two or more things at the same time, as it breaks the monotonicity.

Perhaps you will want to hold off studying calc III until you've done some linear algebra. You don't have to, though.

Also, how well do you know matrices?? If you don't know anything about matrices, then my recommended book will probably be a little too difficult. If you don't know about matrices, then perhaps you should watch the khan academy videos first. You should be very comfortable to

- adding and multiplying matrices
- inverting matrices
- determinants (of 2x2 and 3x3 matrices at least)
- solving systems of equations using matrices

If you find my recommended book too difficult, then a Schaum's outline might be a good help.

 

[woundedtiger4]

- adding and multiplying matrices
- inverting matrices
- determinants (of 2x2 and 3x3 matrices at least)
- solving systems of equations using matrices

If you find my recommended book too difficult, then a Schaum's outline might be a good help.

Yeah, I know this level of matrices. Though I haven't learned it from Khanacademy actually I learn quickly by reading not by watching tutorials and also it consumes lot of time.

will it be OK if I study linear algebra friedberg 2nd edition as I can't find the 4th edition on internet to download :(

 

[micromass]

Any edition would be fine.

And downloading books is illegal.

 

[woundedtiger4]

Any edition would be fine.

And downloading books is illegal.

Excellent.

I am ready to pay if it's available in pdf format because then if I don't understand any term or any specific piece I just copy/paste it in google & it saves time.

 

[woundedtiger4]

.

Hi! I have sent you a pm, please reply me whenever you are free.

 

[rjb]

Depends on what you want to do later. If you want to do physics, then you don't necessarily need Spivak.
If you're into math, then you'll need to study a rigorous calc/analysis book sooner or later. And Spivak is a very good book.

I agree with this 100%. Like the originator, I was self taught at first and struggled with analysis in general and Spivak in particular. Later, on going to university as a very mature student to do mathematical physics, I found it necessary to study from "how to do" books at first whereas Spivak is more a "why we do it" book. Both types of book are necessary, I believe for proper understanding and ability to do maths, but you have to study in the right order. You don't start studying harmony etc etc before you can play a note if you are studying a musical instrument - you learn to play a little bit and then learn theory and practice in tandem ie Stewart and Spivak in parallel - though I personally find Anton hard to beat for calculus; there are several good analysis books though I would have to think a bit about these - have you seen Apostol's book on analysis - very good, even if published years ago

 

[mathwonk]

here's a free linear algebra book, and only 15 pages long. ("you get what you pay for".)

 

[mathwonk]

here are some linear algebra books by friedberg, insel and spence for $1. In this case you may well get more than you pay for.

http://www.abebooks.com/servlet/SearchResults?an=friedberg,+insel,+spence&sts=t&x=51&y=6

 

[woundedtiger4]

here are some linear algebra books by friedberg, insel and spence for $1. In this case you may well get more than you pay for.

http://www.abebooks.com/servlet/SearchResults?an=friedberg,+insel,+spence&sts=t&x=51&y=6

thanks for sharing the link