Calculus Reference Textbook Request

[athena810]

Hi,

Background:
I'm a junior self-studying Calculus, and I want to study from a rigorous textbook. I've seen many threads comparing and contrasting the textbooks of Apostol and Spivak. I have access to the online versions of both author's textbooks, but I find it difficult to concentrate when the book is on a screen (must be a mind thing), so I'll probably get one in print.
Anyway, I did read the first two chapters of Spivak's book and I found the problems impossible. I think mostly it was because I do not know how proofs work. Frankly, I was confused as to what was being asked. I hear the problems in Apostol's book are easier so i was considering reading that, but I have diffiuclty concentrating, and I think it'll be worst since I hear he's very dry. I do have a textbook from school (Stewart) and I like the pictures, but I probably want to read something more rigorous.

So like I said, I plan to self-study Calculus, because I like it and my teacher isn't really that good, so I'm worried I'll do poorly on the AP (next year). She also does the "memorize and repeat" method of teaching, which really annoys me. I should probably read How to Prove It since I heard that it helps. So, should I learn Calculus before reading Apostol or Spivak, or should I just go for it..?

Thanks

 

[micromass]

Is this your first encounter with calculus? If it is, then Spivak will likely be too difficult for you. It's more of an intro analysis for people who already had some first (possibly nonrigorous) encounter with calculus.

Apostol is a very good book. I don't consider him dry at all, but I'm used to rigorous math books, so I understand how others don't like it. You should give it a try, perhaps read the first two chapters and see how you like it.

Take a look at Lang's "first course in calculus" too. It's also rigorous, but less so than Apostol and surely less than Spivak. It's the right rigor for a first course in my opinion.

In any case, stay as far away from Stewart, Larson and other such books as you can.

 

[athena810]

Is this your first encounter with calculus? If it is, then Spivak will likely be too difficult for you. It's more of an intro analysis for people who already had some first (possibly nonrigorous) encounter with calculus.

Apostol is a very good book. I don't consider him dry at all, but I'm used to rigorous math books, so I understand how others don't like it. You should give it a try, perhaps read the first two chapters and see how you like it.

Take a look at Lang's "first course in calculus" too. It's also rigorous, but less so than Apostol and surely less than Spivak. It's the right rigor for a first course in my opinion.

In any case, stay as far away from Stewart, Larson and other such books as you can.

Ok, and yeah i found Spivak sort of confusing; he doesn't state the obvious, but I hear Apostol does, which is good.
Just wondering, why is Stewart's book considered not a great book? It seems like a pretty standard Math textbook.

 

[IGU]

Just wondering, why is Stewart's book considered not a great book? It seems like a pretty standard Math textbook.

Stewart is pretty standard. Great is a whole different thing. Stewart's not rigorous, generally doesn't prove things. Not for mathematicians.

 

[DrewD]

Ok, and yeah i found Spivak sort of confusing; he doesn't state the obvious, but I hear Apostol does, which is good.
Just wondering, why is Stewart's book considered not a great book? It seems like a pretty standard Math textbook.

Stewart is pretty standard. Great is a whole different thing. Stewart's not rigorous, generally doesn't prove things. Not for mathematicians.

I personally think Stewart is an excellent book for somebody looking to learn calculus to use calculus. The text is not completely rigorous, but rigor doesn't necessarily help you understand the subject. My problem with Stewart is the damn price of the book! A book that every freshman (in college) has to buy that won't be used again by the math students should cost less since the author will still make a killing.

Anyway, you seem to be getting stuck between two different questions. Do you want "the" calculus book that will teach you how to prove the theorems that you use and give you a deep understanding of the rigorous logic, or do you want to learn to use calculus for the AP?

A book like Stewart will be a lot better for self study for the AP than Spivak. I also don't know Apostol, but a quick look on Amazon make it seem similar to Spivak in the sense that proofs and rigor are important. This will not help you on the AP. Sure, if you read the entire book and understand it all, you will do well on the AP.

Personally, I would look for Stewarts book at your library and see what you think. I would study from that book (or a cheaper similar book) and use the digital copies of Spivak or Apostol to supplement the sections that you don't feel are rigorous enough. Otherwise you risk getting bogged down proving the MVT and not learning the things that are important for the AP.

PS How to Prove It is a good book to help you through your first proof book, but it will also be useless for the AP part of your goal.

 

[athena810]

Anyway, you seem to be getting stuck between two different questions. Do you want "the" calculus book that will teach you how to prove the theorems that you use and give you a deep understanding of the rigorous logic, or do you want to learn to use calculus for the AP?

A book like Stewart will be a lot better for self study for the AP than Spivak. I also don't know Apostol, but a quick look on Amazon make it seem similar to Spivak in the sense that proofs and rigor are important. This will not help you on the AP. Sure, if you read the entire book and understand it all, you will do well on the AP.

Personally, I would look for Stewarts book at your library and see what you think. I would study from that book (or a cheaper similar book) and use the digital copies of Spivak or Apostol to supplement the sections that you don't feel are rigorous enough. Otherwise you risk getting bogged down proving the MVT and not learning the things that are important for the AP.

PS How to Prove It is a good book to help you through your first proof book, but it will also be useless for the AP part of your goal.

I think my goal is to really understand Calculus and not just memorize things. The understanding part of math is the reason why I like math at all, and I don't want to just memorize equations and theorems and not understand how they work. I'm not sure if this would be considered rigor or if I'm looking for rigor.

I was never good at geometry proofs, so I'm not sure how I'll do with these proofs. I started reading How to Prove It, but it was kinda boring...(like I said, I have problems concentrating). I'll probably take math in university and take real anaylsis then, except I really want to understand Calculus as I learn it.

 

[micromass]

I think my goal is to really understand Calculus and not just memorize things. The understanding part of math is the reason why I like math at all, and I don't want to just memorize equations and theorems and not understand how they work. I'm not sure if this would be considered rigor or if I'm looking for rigor.

I was never good at geometry proofs, so I'm not sure how I'll do with these proofs. I started reading How to Prove It, but it was kinda boring...(like I said, I have problems concentrating). I'll probably take math in university and take real anaylsis then, except I really want to understand Calculus as I learn it.

I was never good at geometry proofs either as they are done in high school. Proofs in calculus tend to be quite different.
I also find "How to Prove It" pretty boring. There is a lot of material in there that you won't need for calculus.

If you're into rigor, then Stewart is not at all for you. It might be a standard textbook, but that doesn't make it good (certainly not for math majors!). Try Lang or Apostol, or maybe even Lax.

 

[jbunniii]

I think my goal is to really understand Calculus and not just memorize things. The understanding part of math is the reason why I like math at all, and I don't want to just memorize equations and theorems and not understand how they work. I'm not sure if this would be considered rigor or if I'm looking for rigor.

I was never good at geometry proofs, so I'm not sure how I'll do with these proofs. I started reading How to Prove It, but it was kinda boring...(like I said, I have problems concentrating). I'll probably take math in university and take real anaylsis then, except I really want to understand Calculus as I learn it.

I agree with micromass's suggestion - get Serge Lang's "First Course in Calculus". This book is rigorous enough to give you a good understanding of why the results are true, without getting too bogged down in highly technical proofs. It is a very nice middle ground between plug/chug texts like Stewart vs. "baby real analysis" texts like Apostol and Spivak. The latter two books are outstanding, but they are more suitable for a second exposure to calculus, in my opinion.

 

[verty]

My prime recommendation is for this first book, I haven't read it but it seems to be one of the more highly regarded sub-honors books. What you may want to do is to match it with a rigorous book that covers the material in roughly the same order, so you can read them in parallel and get the accessibility and problems as well as the theory. I'll be a little different and recommend the second book below as a good and cheap companion text for this purpose. I think everyone needs a challenge to stay interested and the shortest path is the direct one so this small book is likely a good choice.

https://www.amazon.com/dp/0201531747/?tag=pfamazon01-20

https://www.amazon.com/dp/0486650383/?tag=pfamazon01-20

From that book:

The object is to redo calculus correctly in a setting of sufficient generality to provide a reasonable foundation for advanced work in various branches of analysis. The emphasis is on abstraction, completeness and simplicity.

Disclaimer: These are not reference books but they are what I recommend to an ambitious junior in high school who wants a very solid understanding of calculus.

 

[DrewD]

I think my goal is to really understand Calculus and not just memorize things. The understanding part of math is the reason why I like math at all, and I don't want to just memorize equations and theorems and not understand how they work. I'm not sure if this would be considered rigor or if I'm looking for rigor.

I was never good at geometry proofs, so I'm not sure how I'll do with these proofs. I started reading How to Prove It, but it was kinda boring...(like I said, I have problems concentrating). I'll probably take math in university and take real anaylsis then, except I really want to understand Calculus as I learn it.

The questions is whether you want to understand it from a mathematician's standpoint or a scientist's* standpoint. For the former, Stewart is not good, but neither is the AP. For the latter, in my opinion, Stewart or similar is okay (and free since it is in your school library) and it is more closely related to the type of questions on the AP. Remember, neither Newton nor Leibniz understood calculus in a way that would be considered rigorous today.

Again, I think "understanding calculus" does not have one meaning. I have a number of friends that are physicists that I would say "understand" calculus, but can't prove many of the basic theorems and don't care about the construction of the real numbers etc.

I like verty's idea. Use a less scientist's calc book along with a rigorous book on the side. With only a rigorous book you run the risk of getting caught up understanding the intricacies of specific proofs (like MVT) when just recognizing that the MVT follows quickly from Rolle's Theorem is enough. I had no problem with Stewart (I may be the only person on Earth that feels this way). You could tell that he was leaving a lot out, but I got a nice quick understanding of calculus and then took a topology course (and breezed through analysis after almost dying in topology).

If you are truly dedicated to spending the time to learn calculus rigorously, that is awesome, go for it. But I have had a number of students, after learning precalc which is basically just memorizing a bunch of things to prepare for calculus, think they wanted to learn real calculus only to get overwhelmed and decide that they were not interested in math at all. I don't know you, but I was happy to get a non-rigorous overview (enough to get a BS in physics) before taking the more rigorous course.


*This may not be the best choice of terms. What I mean is someone who is concerned with the application of mathematical ideas to the real world rather than somebody who is concerned with the axiomatic construction of mathematics.

 

[verty]

Again, I think "understanding calculus" does not have one meaning. I have a number of friends that are physicists that I would say "understand" calculus, but can't prove many of the basic theorems and don't care about the construction of the real numbers etc.

But you have said that Stewart leaves out a lot. You said that it didn't matter because you only wanted to use calculus, not understand it. But Athena810 has said/intimated that he/she wants to understand it, not just use it. So that type of book is not what Athena810 wants.

Also, Strang's Calculus text is available brand new for $80 and is available freely online from MIT. So if one did for some reason want a plug 'n chug book (I personally can't learn from such books), why not choose that one instead?

 

[DrewD]

But you have said that Stewart leaves out a lot. You said that it didn't matter because you only wanted to use calculus, not understand it. But Athena810 has said/intimated that he/she wants to understand it, not just use it. So that type of book is not what Athena810 wants.

Also, Strang's Calculus text is available brand new for $80 and is available freely online from MIT. So if one did for some reason want a plug 'n chug book (I personally can't learn from such books), why not choose that one instead?

I think you misunderstood what I said; I did want to understand calculus. Stewart and others like the Finney book you posted do allow you to understand calculus, just not at the level necessary for a degree in mathematics. I don't think that a 17yo (give or take) will be hurt by Stewart in conjunction with an analysis book. One of my professors was self-taught from books like Baby Rudin because he grew up poor in Haiti, but even he didn't recommend that style of education.

In my last post I recommended that Athena810 use the copy of Stewart in his/her library for the intuitive understanding of calculus and purchase (or find another library with) a copy of a more rigorous text to supplement Stewart's presentation which ignores topology and some of the Cauchy style of rigorous calculus. If Athena810 means rigorous in that way, then you are correct, but in my experience, most students don't know what rigorous means to a mathematician. You can still understand a lot with pre-1800's calculus.

Athena810, don't take my recommendation as discouraging you from tackling the more rigorous texts. Just don't be discouraged if you find Spivak difficult. Remember that it is a book that was written for an honors calculus course where the students were probably only taking 3 or 4 classes and had a teacher/TA that was actually using that text and the student very well may have already taken calculus in HS, probably using a less rigorous text.

 

[verty]

I think you misunderstood what I said; I did want to understand calculus.

Let me be accurate. You said, "I personally think Stewart is an excellent book for somebody looking to learn calculus to use calculus.". You reinforced this with "or do you want to learn to use calculus for the AP?". Suppose now that you wanted understanding and got it from Stewart. Wouldn't you have said, "use Stewart for understanding"? Why was the loud and clear message "learn to use"?

Something different now. What is misunderstanding? It is having the wrong conception or wrong belief. Is a coincidental belief understanding? If I happen to believe what is true, is that understanding? If I learn a formula from a book, I haven't checked it, I haven't seen it proved, there could be typo's, there could be corner cases, but supposing it happens to be the truth, is this understanding?

 

[DrewD]

Let me be accurate. You said, "I personally think Stewart is an excellent book for somebody looking to learn calculus to use calculus.". You reinforced this with "or do you want to learn to use calculus for the AP?". Suppose now that you wanted understanding and got it from Stewart. Wouldn't you have said, "use Stewart for understanding"? Why was the loud and clear message "learn to use"?

I meant to make my message "People may mean different things by 'understanding'". That was clear to me, but I guess I didn't convey it very well, sorry.

Something different now. What is misunderstanding? It is having the wrong conception or wrong belief. Is a coincidental belief understanding? If I happen to believe what is true, is that understanding? If I learn a formula from a book, I haven't checked it, I haven't seen it proved, there could be typo's, there could be corner cases, but supposing it happens to be the truth, is this understanding?

I absolutely agree that being right does not imply understanding. But I also think that it is unfair to give students the idea that one needs to be able to prove all of the major theorems to understand calculus. IVT and EVT were not proved until the 19th century. Stewart does not prove these. He does prove (a version of) the Fundamental Theorem and MVT. He doesn't fully prove the chain rule in the main flow of the text, but provides an appendix (admittedly, he doesn't do it well). IMO we do a disservice to students if we give them the impression that they don't understand calculus without knowing the theorems that were just assumed to be true until Bolzano and Cauchy came around. Certainly, one needs to learn about these topics as soon as possible, but even very intelligent students sometimes find theorems like Bolzano-Weierstrass and Heine-Borel to confuse the meaning of otherwise simple ideas of calculus.

 

[micromass]

IMO we do a disservice to students if we give them the impression that they don't understand calculus without knowing the theorems that were just assumed to be true until Bolzano and Cauchy came around.

I agree. It's certainly true that you don't need a rigorous account of calculus in order to understand it. But the OP specifically said he was interested in a rigorous text. In that case, it's hard to recommend Stewart.

Lang's text also doesn't provide rigorous epsilon-delta proofs for the intermediate value theorem. I think it would be a mistake for somebody new to calculus to spend much time learning such proofs. But the book is way more rigorous than Stewart, and I think it would suit the OP well.

 

[jbunniii]

Lang's text also doesn't provide rigorous epsilon-delta proofs for the intermediate value theorem. I think it would be a mistake for somebody new to calculus to spend much time learning such proofs. But the book is way more rigorous than Stewart, and I think it would suit the OP well.

It's true that Lang deliberately avoids epsilon-delta proofs in the main text (indeed, he says in the foreword, "My opinion is that epsilon-delta should be entirely left out of ordinary calculus courses"), but he does in fact have an appendix on $\epsilon$-$\delta$ in which he provides proofs of the EVT and IVT: theorems 4.2 and 4.3, respectively, in appendix 4 in my edition.

He writes a few paragraphs at the start of the appendix, encouraging most readers to skip it unless they are theoretically inclined. In the main part of the text he simply states these theorems, and says that they are intuitively clear but proved in the optional appendix. Seems like a good compromise to me for a first course.

 

[micromass]

It's true that Lang deliberately avoids epsilon-delta proofs in the main text (indeed, he says in the foreword, "My opinion is that epsilon-delta should be entirely left out of ordinary calculus courses"), but he does in fact have an appendix on $\epsilon$-$\delta$ in which he provides proofs of the EVT and IVT: theorems 4.2 and 4.3, respectively, in appendix 4 in my edition.

He writes a few paragraphs at the start of the appendix, encouraging most readers to skip it unless they are theoretically inclined. In the main part of the text he simply states these theorems, and says that they are intuitively clear but proved in the optional appendix. Seems like a good compromise to me for a first course.

Yes. But to be fair, his appendix is pretty horribly written. I wish he spent some more time on it to actually explain the intuition behind it. As it stands, I don't think many people will find it useful. The book is still awesome though, but there are much better books to learn epsilon-delta stuff from than from his appendix.

 

[athena810]

I'm looking through Stewart's text and he does prove a bunch of things...but I don't know if you mathematicians consider that rigorous.

I did find Lang's 4th Edition A First Course in Calculus and I like how he explains things. It's very clear. The only thing is that, I do know a lot of what he's writting already, so the majority is nothing new, but he does clarify a few concepts that I did not understand before, which is great.
I think I was a little misleading in my first post; I do know basic differentiation and integration. As in, I know how they work and I know how to carry out the operations. I mostly learned this by wathcing khanacademy videos...so no formal education there. Sorry, I forgot to include that part.

 

[jbunniii]

Hmm, so it sounds like you are looking for a book that is more theoretical than Lang's but easier than Spivak's. I'm not sure what would be the best choice, but you mentioned Apostol as possibly fitting this category. I would say that Apostol is just as challenging as Spivak, and maybe even a bit more so because his exposition is not quite as detailed (I think Spivak tends to show more intermediate steps in his proofs, for example). Apostol's problems might be slightly easier on average, but you will still have a hard time with them if you are new to proofs.

Maybe take a look at this book by Ross: https://www.amazon.com/dp/1461462703/?tag=pfamazon01-20 I have not read this book, but my impression is that it's an easier version of Spivak, more or less. Certainly the exercises look easier, based on the Amazon preview...

Another option would be to stick with Spivak and post in the HW forums here whenever you get stuck or if you just want to check your solutions.

 

[xristy]

Courant and John - Introduction to Calculus and Analysis

Take a look at Courant and John - Introduction to Calculus and Analysis. The first volume is single variable calculus with an analytic flavor and a many applications. It can be thought of as between Lang and Spivak.

 

[finnk]

for your purpose, imo, the best way to study for ap is to get those ap review book, such as princeton, and supplement it with schaum's if you want more.

if you want to understand calculus read stewart (because you already have it, what's the use of having a book if you never read it ?), it's good for developing intuitive understanding of the subject. read analysis by its history by e hairer and g wanner on the side. and perhaps the calculus gallery by william dunham.

now, if you're interested in mathematics as a subject, here's my recommendation,

how to think like a mathematician by kevin houston
introduction to analysis by arthur mattuck

for your first introduction to proof. both are great book, with excellent advice. and you really can't go wrong with the price.

other area of interest, that is accessible for high school student,

vector calculus, linear algebra, and differential forms by john hubbard and barbara hubbard
contemporary abstract algebra by joseph a gallian
visual complex analysis by tristan needham
introduction to topology by colin adams and robert franzosa
the knot book by colin adams
a walk through combinatorics by miklos bona
elementary number theory by david m burton
concrete mathematics by ronald graham, donald knuth, and oren patashnik

the above books aren't the most rigorous books on the subject. don't worry about it, it will come with time and experience.

i'm going to suggest you skip reading courant, apostol, or spivak for now.