Calculus Textbook Recommendation (for Chemist/Microbiologist)

[bacte2013]

Hello!

I am an undergraduate student majoring in the chemistry and microbiology. Summer vacation is truly a great time to teach myself some some subjects besides conducting the scientific research, and I am decided to self-study the calculus over this summer (I took the introductory calculus but the course was not too deep but focused only on the general problem-solving). I am planning to purchase some "deep, proof-based, and rigorous" calculus textbooks because I realized that physical chemistry, quantum mechanics, and physics (and chemistry itself) that I have to take later on really depend on the proof-based calculus. I am also thinking of purchasing one or two books from Spivak, Courant, Lang, or Apostol (I like to buy all of them but each costs a lot). What calculus textbook from above do you recommend to me? I am planning to finish the book by the end of the summer vacation. If you also know a calculus textbook that is dedicated to the chemistry, please let me share your information too.

Have a wonderful day!

 

[ghostwind]

How much calculus did you take and what textbook did you use in school?

 

[verty]

Lang, you are likely to find cheap versions of it available because his books are very popular. Also I think you don't really need the FULL rigor of calculus for chemistry.

 

[bacte2013]

I took the single-variable calculus course on the Fall; we used the book Stewart Calculus, which was okay book not not that deep and intuitive. Basically, the course was a computational calculus. I heard good reputation about the Lang, but I heard that it is not as good as Spivak or Apostol. In order to understand the derivations inside the physical chemistry and quantum mechanics, I think the full-rigor calculus is needed. However, please correct me if you think I am wrong on both Lang and mu opinion. Is there a good calculus textbook dedicated to the chemistry?

 

[mathwonk]

here is a free copy of courant vol 1:

https://archive.org/details/DifferentialIntegralCalculusVolI

and a free copy of apostol:vol1:

http://www.matematica.net/portal/e-books/Apostol%20-%20CALCULUS%20-%20VOLUME%201%20-%20One-Variable%20Calculus,%20with%20an%20Introduction%20to%20Linear%20Algebra.pdf

 

[bacte2013]

Thank you very much for the link, mathwonk! I think I will go with Spivak, as I did not like Courant and Apostol that much. However, I heard that Spivak only covers the single-variable, and he also left out the differential equations. Could you recommend a multi-variable calculus + differential equations textbook that I can study after Spivak?

 

[verty]

Courant actually looks very similar to Spivak, it's almost like Spivak set out to write a book in the style of Courant. So I would make a serious attempt at doing all problems in Courant, this is going to be a very similar experience to any book of this sort (but Spivak surely has more challenging/interesting problems).

 

[bacte2013]

OMG! I just looked at the Spivak's Calculus (4th edition), and the book was tough! Should I just stick to Lang's A First Course in Calculus/Stewart? Do you think Spivak will be helpful for the chemist or microbiologist?

 

[TheAustrian]

Spivak will be too rigorous for Chemistry. You won't need to know an awful lot of maths if you're just doing your first 4 years.

 

[finnk]

In order to understand the derivations inside the physical chemistry and quantum mechanics, I think the full-rigor calculus is needed.

I don't think full-rigor calculus is necessary. Its nice but not necessities. Book like Mary L Boas Mathematical Methods in The Physical Sciences or similar should cover everything you need to know.

 

[ghostwind]

OP, take a look at this thread I made:

https://www.physicsforums.com/showthread.php?t=758357

I believe the problem is again with the Stewarts and similar types of computational books, which seem to have been watered down with each new edition. Intelligent students will want more - they will know something is missing. But they may not necessarily need or be ready for the rigor of Courant, Spivak, or Apostol. Their background may be insufficient for that level just now, but not insufficient for what I called in my thread linked above, "the intermediate" books category. These are calculus books by George Simmons, Serge Lang, etc. These from what I've seen, strike a good balance between application/computation and theory, proofs, etc. And I believe, if adequately prepared prior to university, anybody can use Simmons or Lang. But universities in the US (seeing as high school students are not very good at precalculus even), have to play to the lowest common denominator and thus use books like Stewart, Thomas, etc. If people are struggling with Stewart, how can they use a Lang or a Simmons, never mind Spivak, Apostol, Courant. If the US can somehow get its high school math curriculum in order, and better prepare students for calculus (maybe not even OFFER it in high school as one idea and instead better prepare them for calculus), then maybe, maybe they can offer books like Lang or Simmons. MIT uses Simmons for its regular, non-honors calculus and has for years. For their theory/honors (like Caltech), they use Apostol. But they assume students for those courses will have already been exposed to a good does of quality calculus in high school. And MIT and Caltech are the TOP schools in the US. They will attract the best of the best. The others? They need to keep it watered down unfortunately...

 

[TheAustrian]

I can't imagine High School without Calculus. Might as well then withdraw the math curriculum from HS.

 

[ghostwind]

I can't imagine High School without Calculus. Might as well then withdraw the math curriculum from HS.

It's a sad state of affairs in the US. I wasn't original in my thought. I had heard that argument before being made. A thread on this forum here:

https://www.physicsforums.com/showthread.php?t=312799

Maybe the focus in fixing the system needs to be first in quality not quantity. Then focus on quantity if you want to catch up with the rest of the world.

 

[TheAustrian]

It's a sad state of affairs in the US. I wasn't original in my thought. I had heard that argument before being made. A thread on this forum here:

https://www.physicsforums.com/showthread.php?t=312799

Maybe the focus in fixing the system needs to be first in quality not quantity. Then focus on quantity if you want to catch up with the rest of the world.

Wow. I'm shocked by this information. I assumed that Calculus was unilaterally a standard thing everywhere in the last (two) year(s) of high school. Actually I'm confused as hell about US maths education. What topics do they cover in HS?

 

[bacte2013]

So the calculus level of Spivak or Lang is not required for the chemistry and biology?

 

[TheAustrian]

No, not at all.

 

[verty]

According to Micromass, Lang is not that rigorous. And looking at Lang's second book, it looks pretty much like any other multivariable book so Lang may be a good middle ground between the usual books and books like Spivak.

That said, you started off talking about proof-based course that will come later, so I still think Courant's volume 1 looks very good, you should give it a good go, I think it can work out. Spivak is as hard as nails, this is well known and is sort of why people choose it, so don't let that put you off other books that won't be quite the same.

 

[bacte2013]

Thank you very much for the suggestion! Do you know the calculus textbooks related to the chemist? I looked at the Mathematical Methods for Physical Sciences by Boas, but the book mainly focuses on the physics, which is not my purpose. I searched the book and found the book called ""The Chemistry Maths Book" by Erich Steiner (in Amazon). DO you think this is a good book?

 

[bacte2013]

Thank you very much for all the helpful information and suggestion! I made up my mind and I think I will spent the summer with Lang's A First Course in Calculus and Schaum's Calculus series. Do you think it is a good idea to read Spivak for deeper understanding (but not solving the problems in that book) after reading corresponding chapters in Lang?

 

[verty]

I think you have found the books you wanted so now it is up to you. I think Spivak is for a later time when you know calculus and want to try to solve those challenging problems.

 

[ghostwind]

According to Micromass, Lang is not that rigorous. And looking at Lang's second book, it looks pretty much like any other multivariable book so Lang may be a good middle ground between the usual books and books like Spivak.

That said, you started off talking about proof-based course that will come later, so I still think Courant's volume 1 looks very good, you should give it a good go, I think it can work out. Spivak is as hard as nails, this is well known and is sort of why people choose it, so don't let that put you off other books that won't be quite the same.

After myself asking somewhat similar questions, and looking through these books, I agree with this. I put Lang in the "intermediate" textbook category, as I was saying above in my post, along with Simmons, which is not discussed here very much, but extremely well written. Having looked at both this past week, I think Simmons is the better writer, but Lang is very good too, though he seems in a rush at times. Either will do just fine as a step before Apostol, Spivak, etc., *IF* you find that you want to go that route afterwards.

 

[bacte2013]

Once again, thank you very much for all the helps! I just bought the Lang's A First Course in Calculus, Lang's Multivariable Calculus, Spivak's Calculus, and Schaum's Calculus and 3000 Problems books (actually I think I will return Schaum's books). Lang and Spivak are perfect for me! For the people who took physical chemistry and quantum & statistical mechanics, does Spivak or Apostol help for the mathematical portions of the chemistry? Also why do publicly-popular introductory calculus textbooks like Stewart and Larson get a lot of criticism? I personally thought that Stewart was a good book in terms of connecting the calculus to the real world...please correct me if I am wrong.

 

[ghostwind]

Wow. I'm shocked by this information. I assumed that Calculus was unilaterally a standard thing everywhere in the last (two) year(s) of high school. Actually I'm confused as hell about US maths education. What topics do they cover in HS?

The answer to your question is, it depends on what HS you go to. To try and create a common curriculum across the country, the Common Core was created, but that has proven to be problematic and I'm not sure helpful. You can read about it and see how the state I'm in (Massachusetts) will do away with it here: http://hechingerreport.org/content/common-common-core-fractures-state-support-falters_16420/

So what's a person to do? I happen to live in a good state, with good public schools. Even then, I have a son in kindergarten, and I most likely will put him in something like this http://www.russianschool.com to supplement, as even the good schools here are weak in their math curriculums from what I've seen and experienced myself as a high school student long time ago.

As for myself, I went to a decent high school here in MA, but I did not take calculus my senior year, even though they offered AP Calc. I didn't because I wanted to take it fresh in college and at a higher level than what my HS offered. So I did well, got into a good Ivy League school, and took all honors calculus 1-3, linear algebra, diff eq, differential geometry, etc. as a physics major. I took most of my courses in the math department, as my school had 3 trains for math - in the math department (hardest), in the engineering department (easier and more applied), and for the non-science or biological science majors, etc. I got As in all of them and loved it, and loved the challenge. And I think if I had taken the calculus AP in HS, it would have been a mistake. I would have come in not very prepared for calculus 2 at Ivy League university honors level for sure.

I did one semester take a math course in the engineering department, and it was not as challenging. I didn't know how it would impact me until the following semester, when I took the a course in mathematical physics and another in differential geometry, and I realized how taking a course that's easier (and used a more computational textbook) was bad. I had to work harder to catch up. So it's important to start off correctly, take the hardest courses, and not to deviate. This is why now, as I plan to go back to finish my degree (I left a semester shy of getting my BA as I was offered a six figure salary to work, and so I put my degree on hold for too long - anyway, that's a long story), I'm looking around and asking here about textbooks. I didn't keep what I used, I forgot and need to review and retake, and in looking at these new books, it seems crazy! I remember my books being smaller in size and more black and white :) Anyway, that's my story, and maybe will give you some insight into the US system a bit.

 

[finnk]

Read the preface of Boas book. I think it is perfect for your purpose. It would be better though, if you manage to take the math class that cover the material in Boas book. There are many alternatives to Boas that I'm not familiar of, try to check them out and see which one you like better. Any book that you can learn from is a good book.

Spivak book is not very high yield for chemistry. Apparently he doesn't care about applications. But his book is very clear, and his problems are nice and fun. Know the material in Boas first, it will help you for your P-Chem class. Then you can go back too Spivak later.

 

[bacte2013]

Thank you very much for the suggestion, Mr. finnk! However, I found the better book than Boas book (I believe Boas book focuses on more of theoretical and general physics). I bought the "The Chemistry Maths Book" by Erich Steiner (Amazon.com). So far, I love the combination of Lang, Spivak, and the book I mentioned here. I think I need more than two months to even read the Spivak without doing all the problems.

By the way, I saw so many calculus textbooks (Kline, Thompson, Simmons, Thomas,Silverman, Larson (?), etc.) in the Amazon. Do I need to read them all in order to gain more comprehensive and different perspective & approach to the field of calculus? Or are Lang and Spivak enough to cover other calculus textbooks? I really do not want to spend my time reading many different calculus textbooks since I have to conduct the research and read the books on my field of interest...

Thank you very much for all of your helps! This is truly the best website to seek the helpful advice!

 

[ghostwind]

By the way, I saw so many calculus textbooks (Kline, Thompson, Simmons, Thomas,Silverman, Larson (?), etc.) in the Amazon. Do I need to read them all in order to gain more comprehensive and different perspective & approach to the field of calculus? Or are Lang and Spivak enough to cover other calculus textbooks? I really do not want to spend my time reading many different calculus textbooks since I have to conduct the research and read the books on my field of interest...

Thank you very much for all of your helps! This is truly the best website to seek the helpful advice!

No, that's overkill! As I was saying, there are 3 main categories of books universities and high schools use -> the 1000+page ones with tons of examples, very computational, new editions every other year, etc. (Thomas, Stewart, etc.), the "intermediate" ones which seem to balance things somewhat (Simmons, Lang, etc.), and the very theoretical/analysis ones (Apostol, Spivak, etc.). I would use one from the Thomas/Stewart/Larson category (for problem solving) + one from the Apostol/Spivak/Courant category (for deeper understanding). Or you can just use an intermediate one and be done with it - Lang or Simmons. Don't go too crazy! IMHO, Simmons' book is the single best and most balanced one.

 

[bacte2013]

Thank you very much, Mr. ghostwind (I not sure if this is a correct way to address you, though). Today's I actuallt registered for the course called "Multi-variable Calculus and Linear Algebra" for my upcoming Fall Semester. I looked at the textbook requirement for that course, and the course requires the Apostol's Calculus (Volume 1 and 2). I have some questions about the multi-variable calculus and linear algebra in general. Series and Sequence are not my strongest point in the calculus; are series and sequence treated in the multi-variable calculus and the linear algebra? I am going to review them with Lang's book, but I am curious about this. Also I reckon Apostol's treatment on linear algebra will be bloody brutal...do you know any linear algebra textbook that I can supplement with Apostol's book, and quite elementary? Also is it okay for me to study calculus and linear algebra simultaneously?

 

[verty]

No, that's overkill! As I was saying, there are 3 main categories of books universities and high schools use -> the 1000+page ones with tons of examples, very computational, new editions every other year, etc. (Thomas, Stewart, etc.), the "intermediate" ones which seem to balance things somewhat (Simmons, Lang, etc.), and the very theoretical/analysis ones (Apostol, Spivak, etc.). I would use one from the Thomas/Stewart/Larson category (for problem solving) + one from the Apostol/Spivak/Courant category (for deeper understanding). Or you can just use an intermediate one and be done with it - Lang or Simmons. Don't go too crazy! IMHO, Simmons' book is the single best and most balanced one.

I don't like anyone to say "book X is the single best book" because it just isn't true. No one has seen every book, you just can't honestly say that. Also and more importantly, every learner is different. What one person likes, another person will dislike. It often happens that one person loves Larson, another despises it, etc. This is pretty normal and it goes to show that there is no best book. Everyone likes what they like and that is usually different.

You, Ghostwind, like Axler and Simmons. Other people will like something else. We can't say what book is the best, that is the point.

 

[verty]

Also I reckon Apostol's treatment on linear algebra will be bloody brutal...do you know any linear algebra textbook that I can supplement with Apostol's book, and quite elementary? Also is it okay for me to study calculus and linear algebra simultaneously?

Yes, you can study them simultaneously, there is almost no overlap. However, linear algebra is usually learned later than calculus and most books will be slightly more advanced than your average calculus book.

I actually think it might be a good idea to go through Lang's calculus book first before moving on to linear algebra.

 

[ghostwind]

I don't like anyone to say "book X is the single best book" because it just isn't true. No one has seen every book, you just can't honestly say that. Also and more importantly, every learner is different. What one person likes, another person will dislike. It often happens that one person loves Larson, another despises it, etc. This is pretty normal and it goes to show that there is no best book. Everyone likes what they like and that is usually different.

You, Ghostwind, like Axler and Simmons. Other people will like something else. We can't say what book is the best, that is the point.

I prefaced it by saying IMHO, so it is an opinion. And from all the intermediate books I've seen, which have been quite many recently, Simmons' stands above them all. Lang is a bit sloppy and rushes through things at times - Simmons is a better writer. I said either is fine though - this is nitpicking.

And as is always the case, each individual is free to make up his or her mind. And each individual is free to say what book he or she may think is best as well. I have no problem with that. I don't see why anyone would!

Have you read Simmons? Lang? Or are you just going on what others have said? Yes, I like Axler, I like Lang, I like Apostol, Simmons, etc. I've surveyed them all in the past few weeks (and many before that over the years). It doesn't take too long to look through them and make an informed opinion, which is what I did. I wish more people would chime in with their own personal informed opinions instead of shying away for no reason.

There is a reason MIT has used and is using Simmons, just as they use Apostol. :)

 

[bacte2013]

Thank you, Mr. verty! My plan is to finish Lang's A First Course in Calculus and Spivak's Calculus over this summer. My next mathematics course is a two-semester course called "Multivariable Calculus and Linear Algebra", which requires the second volume of Apostol. I am planning to purchase Lang's Multivariable Calculus and use it as a supplement to the APostol since Spivak only covers the single-variable calculus. I looked at Simmons' Calculus and Analytical Geometry, but I feel like it is not as good as Lang's. Do I need to purchase Apostol's first volume (single-variable) if I already have Spivak? Do I need to complete the first volume of Apostol in order to understand the second volume of his calculus textbook? Does anyone read the book called "How to Prove It" by Daniel Velleman? I am also planning to purchase the linear algebra textbook(s) to supplement the linear algebra portion of my course (I mentioned that the required textbook is Apostol's second volume)...does anyone know a good linear algebra that is easy to read and not rigorous as Apostol? I saw the linear algebra book wrote by Serge Lang, but I do not know if that is a good book..

I sincerely apologize for keep asking many questions regarding to the mathematics textbook, but I want to make sure I bought the right textbooks.

 

[ghostwind]

Also is it okay for me to study calculus and linear algebra simultaneously?

It's perfectly fine as long as you have Calc I-II under your belt. I took Calc III and Linear Algebra honors in the same semester back in 1993, and was no issue. My issue now is finding good books for reviewing all I forgot :)

 

[verty]

Here are four linear algebra books roughly in order of difficulty. Strang is a book focused on computations, how to calculate things. Lay is probably the closest to your request for an elementary book. The last two are what I call typical books, being more advanced. Lang looks possibly more advanced than these so it is probably not any easier than Apostol would be. Shilov is very clear and is the type of book that you could read without doing problems and still learn a lot and it costs about $10. Shilov with Lay or Strang would be a supremely good introduction, I think. Cullen on its own would also work very well, I believe. But I leave choice to you (or to see what others recommend, of course).

https://www.amazon.com/dp/0155510053/?tag=pfamazon01-20
https://www.amazon.com/dp/0201709708/?tag=pfamazon01-20
https://www.amazon.com/dp/0486663280/?tag=pfamazon01-20
https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20

 

[ghostwind]

Thank you, Mr. verty! My plan is to finish Lang's A First Course in Calculus and Spivak's Calculus over this summer. My next mathematics course is a two-semester course called "Multivariable Calculus and Linear Algebra", which requires the second volume of Apostol. I am planning to purchase Lang's Multivariable Calculus and use it as a supplement to the APostol since Spivak only covers the single-variable calculus. I looked at Simmons' Calculus and Analytical Geometry, but I feel like it is not as good as Lang's. Do I need to purchase Apostol's first volume (single-variable) if I already have Spivak? Do I need to complete the first volume of Apostol in order to understand the second volume of his calculus textbook? Does anyone read the book called "How to Prove It" by Daniel Velleman? I am also planning to purchase the linear algebra textbook(s) to supplement the linear algebra portion of my course (I mentioned that the required textbook is Apostol's second volume)...does anyone know a good linear algebra that is easy to read and not rigorous as Apostol? I saw the linear algebra book wrote by Serge Lang, but I do not know if that is a good book..

I sincerely apologize for keep asking many questions regarding to the mathematics textbook, but I want to make sure I bought the right textbooks.

Whew! One busy summer! So let me try to summarize at least what I understand so far, and what I would do in your situation given I understand it correctly:

1. You have taken Calculus I & II using Stewart's book, which you were fine with, but felt it/the course wasn't as rigorous or in-depth as you'd like and think you will need for further studies.

2. You are looking for a good book to self-study/review Calc I & II from over the summer in a more in-depth/proof-based way than you did in school with Stewart.

3. You looked at Spivak and were surprised to see how tough it was.

4. You decided to use Lang's "A First Course in Calculus" then, but also got Spivak even though you admitted it will take you over 2 months to just read the book, never mind do the problems.

5. You want to ensure you are getting the best book(s) and not missing out on anything.

6. You will be taking a 2 semester course on Linear Algebra & Multivariable Calculus in the fall, which will use Apostol's 2nd volume.

7. You also want to get a head start on linear algebra this summer as well.

This is a summary of what I got out of your posts thus far - the facts as it were.

If these "facts" are true, then what I would say (and actually do myself if in your shoes), is get Apostol Vol. 1, and use that to review and go more in-depth for Calc I & II and to get an intro to linear algebra (the last 4 chapters are linear algebra in Vol. 1). Why complicate things with Lang, Spivak, etc.? You may like those, and they are good books, but if you are going into a course that will be using Apostol Vol. 2 for more linear algebra and multivariable calculus, then why not get familiar with Apostol by reviewing single-variable calculus and intro linear algebra with Apostol Vol. 1? For me, it makes perfect sense to go this route and just stick with Apostol Vol. 1 over the next few months. It will certainly take you pretty in-depth on Calc I & II and introduce you to linear algebra. And you will be perfectly lined up to start your course that uses Vol. 2!

You don't want to go nuts with 5 books over one summer! It would be better I think to get the most out of Apostol Vol. 1 instead of using 5 books. Just my 2 cents and what I would do.

 

[verty]

You don't want to go nuts with 5 books over one summer! It would be better I think to get the most out of Apostol Vol. 1 instead of using 5 books. Just my 2 cents and what I would do.

Don't you think Apostol 1 is too dry for him? It's like HAL explaining calculus. He has backed off the super rigorous books to some degree, I think he has made good choices so far.