"Mathematics" book for revising/studying

[dRic2]

Hi,

I've taken calculus during my engineering degree (that I'm still attending) two years ago. At that time I didn't realize the importance of calculus (and also linear algebra) so I studied it superficially. I think I'm quite good though because I had to use my calculus knowledge in all the other courses, but but I was sticking to the essential minimum of knowledge. Recently I got very interested in physics so I started studying more and this implied revising my calculus skills. I think I improved a lot in calculus/algebra in the last months but I still find it terribly boring (my fault I know). I was wondering if some of you have any suggestion for a book that will help me revise and maybe go further that is kind of "fun" (with examples and applications) so that I can read it/study in the free time. "Calculus for the practical man" is very much what I was looking for but it is a bit under my level of skill. I read I few pages and I don't think I'm going to get much out it.

 

[Wrichik Basu]

I was recommended Apostol's Calculus here. Later, I heard several seniors and professors disapprove of it as overrated, which is why I didn't go through it, though I have both the volumes.

Another book is by Spivak, but I couldn't buy it because of high cost.

 

[dRic2]

Thank you! Did a quick research and I probably prefer the one by Spivak.

but I couldn't buy it because of high cost.

... And that's the problem...

I think I'm going to stick around a little longer

 

[Wrichik Basu]

I think I'm going to stick around a little longer

You should. There are others who can recommend less costly books.

Actually, the books that I follow are not sold online, and are only found in Kolkata, many of which are not printed any more. Hence I cannot recommend those books as you'll not find them anywhere on the net.

 

[Buffu]

I heard several seniors and professors disapprove of it as overrated,

Overrated is a bad criticism, did they put any actual criticism of the book ?

 

[Buffu]

Thank you! Did a quick research and I probably prefer the one by Spivak.

... And that's the problem...

I think I'm going to stick around a little longer

Are you sure you want Spivak ? It has zero practical applications, it is a hardcore maths book. It probably is not helpful if you need calculus for physics. I would say you should go for 2 books by Courant, they contain a lot of examples of practical applications of calculus in physics. It also has a chapter on calculus of variation.

If that is not enough for you then you can opt for any mathematical methods book like one by Mary Boas.

 

[Wrichik Basu]

Are you sure you want Spivak ? It has zero practical applications, it is a hardcore maths book. It probably is not helpful if you need calculus for physics.

I had seen it being brought up in several threads here, so I was curious and wanted to go through the book, and mentioned it to the OP. Since you're saying, I'll give up my wish to buy it.

 

[Buffu]

I had seen it being brought up in several threads here, so I was curious and wanted to go through the book, and mentioned it to the OP. Since you're saying, I'll give up my wish to buy it.

Why ? Did I say anything wrong ?

 

[Wrichik Basu]

Why ? Did I say anything wrong ?

Why are you thinking so?

You said that the book is not good for application in Physics, and I was trying to buy it for that sole purpose, which is why I gave up my intentions.

On the other hand, thanks for the advice

 

[dRic2]

Are you sure you want Spivak ? It has zero practical applications, it is a hardcore maths book. It probably is not helpful if you need calculus for physics.

I wasn't too happy about Spivak, but it is the one I would choose between those two I was recommended by @WrichikBasu. I didn't know whether it had or not practical applications, but I read few pages on pdf and I enjoyed it more than Apostol's Calculus.

So in the end, thank you @Buffu for the new suggestions. I will check them out for sure! :)

 

[Buffu]

I wasn't too happy about Spivak, but it is the one I would choose between those two I was recommended by @WrichikBasu. I didn't know whether it had or not practical applications, but I read few pages on pdf and I enjoyed it more than Apostol's Calculus.

So in the end, thank you @Buffu for the new suggestions. I will check them out for sure! :)

Tell me if you like them or not. I know some more books of this nature.

 

[dRic2]

@Buffu I've downloaded the pdf of Courant vol 1. Judging from the table of contents it looks very interesting (lots of examples)... By the way, while browsing the web I came across this book "Advanced Calculus" by Wood. What do you think about it? I've downloaded that too: less examples (basically zero till the end of the book), but it covers more stuff and the final chapter is called "classical mechanics"... Maybe all the applications are collected there.

 

[Buffu]

@Buffu I've downloaded the pdf of Courant vol 1. Judging from the table of contents it looks very interesting (lots of examples)... By the way, while browsing the web I came across this book "Advanced Calculus" by Wood. What do you think about it? I've downloaded that too: less examples (basically zero till the end of the book), but it covers more stuff and the final chapter is called "classical mechanics"... Maybe all the applications are collected there.

That is Richard Feynman's favorite calculus book. Feynman credits that book for making him integration wizard. While it doesn't cover more advanced topics like multivariable calculus, it covers important tricks like "differentiation under integral sign". You should definitely take a look at it but I won't recommend it as a main book.

 

[Buffu]

There is also The hitchhiker's guide to calculus which is "baby version" of Michael Spivak's Calculus.

 

[MidgetDwarf]

if you find the calculus book too difficult; there is another option. Calculus by Edwin E. Moise. It is between general mass market calculus books of today and Apostol/Spivak. It is closer to Apostol/Spivak. The book explains the Completeness Axiom, Well Ordering Principle, gives a correct proof of Arc Length, shows the power of the Mean Value Theorem and how it connects with almost all the important results of calculus, and quit some more. It is very readable and entertaining read. Can be used as an Intro to Proof book.

If you read Moise, then you could be ready to tackle Rudin, or you know how to apply Calculus to physics problems.

 

[dRic2]

Thank you for all these books' suggestions! It will be really tough to pick one!

ps:
@Buffu

That is Richard Feynman's favorite calculus book. Feynman credits that book for making him integration wizard. While it doesn't cover more advanced topics like multivariable calculus, it covers important tricks like "differentiation under integral sign". You should definitely take a look at it but I won't recommend it as a main book.

why wouldn't you recommend it as a main book? Just curious.

 

[dRic2]

@MidgetDwarf Do you know where to find a preview or the table of contents of the book by Moise? All the review are very very good but it seems to be quite hard to find

 

[MidgetDwarf]

@MidgetDwarf Do you know where to find a preview or the table of contents of the book by Moise? All the review are very very good but it seems to be quite hard to find

I can take a picture and upload it for you. It covers all of the standard topics plus some.

 

[MidgetDwarf]

I can take a picture and upload it for you. It covers all of the standard topics plus some.

The place to find the book. Would be on Amazon.

 

[Buffu]

Thank you for all these books' suggestions! It will be really tough to pick one!

ps:
@Buffuwhy wouldn't you recommend it as a main book? Just curious.

It is really old and like many old books it is difficult to read because topography is bad. Also it doesn't cover multivariable calculus.

 

[vanhees71]

Hi,

I've taken calculus during my engineering degree (that I'm still attending) two years ago. At that time I didn't realize the importance of calculus (and also linear algebra) so I studied it superficially. I think I'm quite good though because I had to use my calculus knowledge in all the other courses, but but I was sticking to the essential minimum of knowledge. Recently I got very interested in physics so I started studying more and this implied revising my calculus skills. I think I improved a lot in calculus/algebra in the last months but I still find it terribly boring (my fault I know). I was wondering if some of you have any suggestion for a book that will help me revise and maybe go further that is kind of "fun" (with examples and applications) so that I can read it/study in the free time. "Calculus for the practical man" is very much what I was looking for but it is a bit under my level of skill. I read I few pages and I don't think I'm going to get much out it.

If you are fascinated by physics and find calculus/algebra boring, the problem might be that you haven't yet seen the right kind of textbook for you. I got hooked to math by the marvelous textbooks "Lectures on Theoretical Physics" by Arnold Sommerfeld, which in my opinion are the best textbooks about classical physics ever written (ok, I hate the ##\mathrm{i} c t## convention he uses in special relativity, but the books were written in the late 1940ies and early 1950ies, what can you do). So, to appreciate masterful use of mathematical methods in theoretical physics, the best I can recommend is vol. 6 on PDEs in this book series.

 

[dRic2]

If you are fascinated by physics and find calculus/algebra boring, the problem might be that you haven't yet seen the right kind of textbook for you. I got hooked to math by the marvelous textbooks "Lectures on Theoretical Physics" by Arnold Sommerfeld, which in my opinion are the best textbooks about classical physics ever written (ok, I hate the ictict\mathrm{i} c t convention he uses in special relativity, but the books were written in the late 1940ies and early 1950ies, what can you do). So, to appreciate masterful use of mathematical methods in theoretical physics, the best I can recommend is vol. 6 on PDEs in this book series.

thank you for the suggestion. I will check that too. My problem is that I want to be sure of what I buy. Usually I go to the library of my university to find a copy of the book I want, but these last suggestions (including yours) doesn't seem to be there. So I'm not completely sure because I don't like to buy a book I know nothing of.

 

[gmax137]

... Calculus by Edwin E. Moise ...

I found myself a copy of this, and just started going through it from page 1. He talks about "open sentence" -- A mathematical statement that can be either true or false depending what values are used.

For example: x + 4 = 7

Unless we know "x," we don't know if "x + 4 = 7" is true or false. So it is "open."

I have never heard "open sentence" before. Is this common terminology? The book is copyrighted 1966; maybe this was trendy at the time?

 

[Buffu]

I found myself a copy of this, and just started going through it from page 1. He talks about "open sentence" -- A mathematical statement that can be either true or false depending what values are used.

For example: x + 4 = 7

Unless we know "x," we don't know if "x + 4 = 7" is true or false. So it is "open."

I have never heard "open sentence" before. Is this common terminology? The book is copyrighted 1966; maybe this was trendy at the time?

I would say it is a common terminology(https://www.mathsisfun.com/definitions/open-sentence.html) but I don't think it matters much in learning calculus.

 

[mathwonk]

you sound rather inexperienced in calculus, and my guess is that both apostol and spivak (and possibly courant) will be too hard for you. i recommend you go back to the university library, read and find a book you like and then go to abebooks.com to find a used copy.

 

[vanhees71]

I found myself a copy of this, and just started going through it from page 1. He talks about "open sentence" -- A mathematical statement that can be either true or false depending what values are used.

For example: x + 4 = 7

Unless we know "x," we don't know if "x + 4 = 7" is true or false. So it is "open."

I have never heard "open sentence" before. Is this common terminology? The book is copyrighted 1966; maybe this was trendy at the time?

Hm, maybe the author wants to confuse his students as much as he can, for whatever reason (maybe he has a trauma and hates students or is simply evil?).

The standard reading is that x+4=7 is an equation with one unknown to be solved. There are three possibilities: (i) it has at least one solution, (ii) it has no solution, (iii) it is undecidable within the used set of axioms. In case (i) you may have two subcases (ia) there's a unique solution, (iib) there is more then one distinct solution.

In this case, interpreting as ##x## standing for any real number we know that this is linear equation with one unknown, which has a unique solution, and indeed subtracting 4 on both sides of the equation tells you that without other possibility you must have x=3. So there's exactly one solution of the equation. Of course, putting for x any other number than 3 leads to a wrong equation.

 

[gmax137]

The standard reading ...

Yes, exactly what I thought. The "open sentence" thing seems to be an unnecessary additional level of abstraction -- I don't see what it buys beyond the "standard reading." That's why I wondered about the vintage, 1966, right in the "new math" era, which to me seemed to be about making simple ideas as abstractly confusing as possible.

As mentioned above, this has nothing to do with calculus and little to do with the OP in this thread; I will let it be and refrain from further distraction.

Thanks!

 

[WWGD]

My suggestion, if you have access to a library , is to browse through the Calc section and try to get a feel for which book is best for you. Spend a few minutes browsing through before making a decision and then buy the one that feels best for you.

 

[Buffu]

Yes, exactly what I thought. The "open sentence" thing seems to be an unnecessary additional level of abstraction -- I don't see what it buys beyond the "standard reading." That's why I wondered about the vintage, 1966, right in the "new math" era, which to me seemed to be about making simple ideas as abstractly confusing as possible.

I won't say it is a redundant term because it is a term from Formal Logic. My guess is, author introduced it so that readers can gain some mathematical maturity. Most math books have a chapter on Logic/Set-theory so readers can write mathematical proofs. Analogous to how physics books have a chapter on mathematical methods.

 

[MidgetDwarf]

Hm, maybe the author wants to confuse his students as much as he can, for whatever reason (maybe he has a trauma and hates students or is simply evil?).

The standard reading is that x+4=7 is an equation with one unknown to be solved. There are three possibilities: (i) it has at least one solution, (ii) it has no solution, (iii) it is undecidable within the used set of axioms. In case (i) you may have two subcases (ia) there's a unique solution, (iib) there is more then one distinct solution.

In this case, interpreting as ##x## standing for any real number we know that this is linear equation with one unknown, which has a unique solution, and indeed subtracting 4 on both sides of the equation tells you that without other possibility you must have x=3. So there's exactly one solution of the equation. Of course, putting for x any other number than 3 leads to a wrong equation.

You have to read it in context. The author explains what it means for something to be "well defined." He further goes on to show that P implies Q, is not the same as Q implies P, etc. Furthermore, he takes this argument further and explains how, if P is false we do not care wether Q is true or false. You know, conditional statements etc...
The author's intent is to show that mathematics is a very precise language, and ambiguity has no place in it!

Keep reading the book. The intro section is weird if you are not used to this thinking. A few pages more and you see what a Field is...

@mathwonk, are you familiar with this book? If you are not, I think you would find it quite interesting and lot to be gained from it. I think its worth having on a book shelf next to Courant.

 

[MidgetDwarf]

Yes, exactly what I thought. The "open sentence" thing seems to be an unnecessary additional level of abstraction -- I don't see what it buys beyond the "standard reading." That's why I wondered about the vintage, 1966, right in the "new math" era, which to me seemed to be about making simple ideas as abstractly confusing as possible.

As mentioned above, this has nothing to do with calculus and little to do with the OP in this thread; I will let it be and refrain from further distraction.

Thanks!

I suggest you look at how he builds integration and diff. It is really beautiful how Moise built it up. Some geometry, explanation of coordinates, what a right hand system is, what the secant to a line is, how we need to redefine a tangent from our experience from Euclidean Geometry, proof by induction, nice and intuitive explanation of the well ordering principle. How the tangent of a circle can be considered a special case of a more general...

Concept of Area. How "Area" is really ambiguous

He even explains how Dedekind cuts relate to two other concepts...

 

[mathwonk]

well i do not have access to the book by moise but i did get to read the preface of the 2nd (and 1st) edition. This is the most clear and persuasive statement of philosophy of teaching and writing a calculus book, plus how to learn that I have seen in any textbook. of course moise is a world class mathematician which always helps. this is not a parallel of spivak or apostol. those are thorough going theoretical books. this book has been written to be useful as both a theoretical and a less theoretical book, by putting the harder theoretical parts at the end of the sections or chapters, so they can be skipped withut losing grasp of the facts to be used. he also tries to convey, even when omitting the theory, both computational techniques and intuitive concepts. If he succeeds in his stated goals, this should be an excellent book. It thus seems to have more in common with courant in its goals and approach. there is however a reason that the other 3 books, apostol, spivak and courant, have been almost univerally recommended for so long. in particular i have never heard anyone call apostol "overrated". anyone who says this would have to go a long way to avoid being considered a very unreliable source in my opinion. although an intelligent person might be able to make a case somehow, i do not know what it would be based on. i would say apostol is a strong candidate for the absolute best calculus book of all, but only for the very strong and serious student. spivak is also superb, and is more "fun" than the somewhat sombre apostol, but spivak is only for the future pure mathematician. apostol also offers more applied ideas, and also does several variables and linear algebra. still i second the opinion that the best way to proceed is to sit in the stacks in the calculus section of a university library and do some reading.

the only book by moise i have read more of is the one on geometry. while very scholarly, rigorous, and correct, i found it unappealing to read for a student. apparently the calculus book was written with more care in readability. but as i said i have not been able to read it, at least not lately. i seem to have looked at it in the past in my career as a teacher, and it apparently did not leave me wanting to own one. maybe that was in the period when i could not afford books, or maybe i just had too many already, or maybe it did not have any content that i myself felt i needed another source for. spivak always has something that i can learn from. e.g. there is a problem that shows how to prove a function with derivative zero is locally constant, without using the MVT, which not everyone knows how to do. someone asked me this question just a few days ago on the professional math site "mathoverflow". perhaps unfortunately, i gave away my copies of both spivak and apostol a few years ago, to the undergraduate math dept library.

If anyone feels benefited by spivak's book, i would like to remind them that he is still alive and earns his living primarily from sales of it.

 

[gmax137]

I suggest you look at how he ...

Thanks! I will keep at it!

 

[vanhees71]

Yes, exactly what I thought. The "open sentence" thing seems to be an unnecessary additional level of abstraction -- I don't see what it buys beyond the "standard reading." That's why I wondered about the vintage, 1966, right in the "new math" era, which to me seemed to be about making simple ideas as abstractly confusing as possible.

As mentioned above, this has nothing to do with calculus and little to do with the OP in this thread; I will let it be and refrain from further distraction.

Thanks!

May be, it's due to the "new math movement" (or however you call this nonsense). Sometimes, I have the impression that the purpose of mathematics and physics didactics is to deform the subject such that even an expert cannot make any sense about it anymore, let alone the poor students who are exposed to these ideas.

At high school we had to use a textbook, where they considered it pedagogically better not to write down the differential in integrals, i.e., instead of writing
##\int \mathrm{d} x x## they wrote ##\int x##. I immediately told the teacher, that I'd not use this nonsensical notation since it's not leading to the correct dimensions to begin with, and that you cannot know wrt. which variable you should integrate. After a short thought the teacher said that I am right and regretted to have to use this strange textbook ;-))).

 

[WWGD]

May be, it's due to the "new math movement" (or however you call this nonsense). Sometimes, I have the impression that the purpose of mathematics and physics didactics is to deform the subject such that even an expert cannot make any sense about it anymore, let alone the poor students who are exposed to these ideas.

At high school we had to use a textbook, where they considered it pedagogically better not to write down the differential in integrals, i.e., instead of writing
##\int \mathrm{d} x x## they wrote ##\int x##. I immediately told the teacher, that I'd not use this nonsensical notation since it's not leading to the correct dimensions to begin with, and that you cannot know wrt. which variable you should integrate. After a short thought the teacher said that I am right and regretted to have to use this strange textbook ;-))).

I guess some of the people who write these books are in bubbles where no one gives them much ( at least negative) input.