Which Calculus Textbooks Are Used At City Tutoring?
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- เผยแพร่เมื่อ 4 ก.พ. 2025
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Delighted to see that these are books you are recommending, my university in Georgia (country!) is using the same books
I am very much delighted to hear a professional mathematician having opinions similar to mine. I am a student of Grade 11. Almost a year ago, I wished to study Calculus and I was recommended textbooks like Stewart's. When I started to read them, I was frustrated with the lack of rigour.
For instance, when I read the part on Functions I could not understand how f(x) can be called a function if its actually the image of x under f for some x in its domain. Also the emphasis on mindless calculations where no one knows what they are doing drove me mad. I just assumed that I am not smart enough to srudy Calculus perhaps because I did know mathematicians like you.
However, Tom Apostol's Calculus Volume 1 changed everything for me. I found out that I was actually inclined towards something called Mathematical rigour which modern textbooks miss. I am still studying it as a hobby and I have reached the Differential Equations.
"... led astray by the glossy pages of mediocrity." Churchillian? Brilliant!
We did indeed cover the Epsilon Delta proof, and it appeared on an exam. Unfortunately, no further discussions of proofs were offered in class. I was fascinated by the precision and realized I must be missing something vital. Ultimately, I discovered discrete math textbooks and videos on proof writing. Understanding proofs is essential to critical thought. It's unbelievable how this was cut out of mathematics.
I can't describe how good i feel being Dr. Strange student
A very intense and much needed talk.
Comfort is the enemy of greatness! Amazing! This sentence should be told more often to students because they (we) live in a fals, upside down world. I am happy that I had that strange feeling after 150h Calc study with Stewart... A strange, shapeless feeling. (We had a conversation about this under the last video.). Now I know what that is! The complete lack of rigour. Instead, Mr Stewart gave us fals hope and fals secure feelings. Very dangerous. I'm switching to Spivak and Apostol. I need challenge and rigour. I want to understand the deepness of Calculus. Thank you Mr. Cromwell! You are one of the very few Professors who sees the world as it is, not as others want it to be. Candyland...
I would certainly recommend you switch to Spivak or Apostol. Thank you for your comment, because it is always a pleasure to read from people who truly take their studies seriously and will not just take algorithms for an answer.
I'm statistician, and i continue doing a lot of mathematics, calculus and linear algebra, and here in spain is the same problem. I have learnt from spivak ( its hard) but apostol is very amazing too. We have created this by educating a minimum effort society with no sense of knowledge like our ancents ued to do
Yes! You are correct about the "minimum effort" society for sure. I spent some time in Spain, by the way (beautiful country!) How are the Math textbooks there for secondary school students? Here in the US, the "new" textbooks are very thick, but very superficial in the content. They are big books because they have more pictures than math, unfortunately.
@citytutoring In the school is more important the tablet tan the book...this is a problem. I've never read a math book without my pencil and paper. When i was young i learnt to solve limits in the school with polinomials approximations(Taylor with the landau notation). Now be happy if boys with 14 years know the significance of a fraction...it is horrible!!!!! Some of The Universities uses for the grade of mathematics books very similar to stewart... a disaster. There is no effort and the level is very poor.
I'm currently a student in the field of theoretical chemical physics, who'd been frankly surprised how poorly all my classmates understood of simple mathematical concepts, followed by physics/physical chemistry during undergraduate studies in chemistry. It kinda began hitting me with an anvil when I began tutoring both high school kids and some juniors in my university. Only thing that actually differentiated our education: the first calculus book I'd read in my life was Spivak.
Got to learn people's hate about it as an upsetting surprise: when some high school kid I was tutoring asked me which books I'd recommend I told him to read Spivak to which he replied that "but I heard it's so hard and it's so unnecessary". Then I realised that they never even realised the actual fact of necessity of rigorous approaches.
And honestly, seeing many people's responses in the comments section on the "applications" of rigorous thinking etc, I can just say that this "unnecessary obfuscation" is the single thing that can make you ask yourself the "why so and not else?" question that is so avoided nowadays... Teaching critical and logical thinking this way is the simplest way to develop one's fluency in all the other fields. The hard thing is to accept learning has to be hard and painful, it's a bitter pill that is going to really shape you as a person.
Thanks to Prof for this beautiful reflection of this idea! I actually sent this video to a couple of my students for them to hopefully learn from it :D
Thank you very much for your support! Yes, you are absolutely correct in your observation. One of the most common "complaints" I get from some sectors of our students (and even their parents, sadly!) is that the approach I defend is "unnecessary" and "over the top". They really do think that school is there to be "fun" (true learning is indeed fun!) but what I mean is that they expect school to be "fun" as if though it were run like a circus with clowns.
I used to think it was just a "young student problem", but then when I met a lot of the parents, I found that was the source of most of the problems in students who are not doing well. Even on this channel, you will read comments by some ignorant people saying that I "roast" students, as if education was about ignoring errors and just all smiles. They have an infantile mentality, and it shows all the time.
I have taught students who started out with ZERO knowledge in Algebra, only to hear them start talking similar to math majors by the end of a year in my class. THEY ARE capable of the fundamentals if they put in the work and pay attention, take notes, and get their act together. Thank YOU for your support and sharing the video! I wish you and your students well.
@ I'm humbled to see such an elaborate response, haha!
I fully agree with what you said, the thing about teaching mathematics is through personal Eureka moments that are usually preceded by a noticeable amount of struggle. Cannot really make it less cumbersome unless you aim for true understanding...
Anyone can learn anything given enough self-discipline, the problem is about having one. It's a thing many adults don't have, let alone students. And it's hard to self discipline yourself for obvious reasons. About the public's attitude to "learning by struggling", I got recommended a very interesting essay a while ago (by I think my supervisor, a quantum physicist who graduated from a top Chinese university), called "A russian teacher in America" by Andrei Toom. It has a good exposition on how someone outside the system first experienced it, and it was one of the best descriptions of "I trade my time for a grade" approach of many students nowadays, unfortunately.
Thanks for writing up the reply! You definitely have another educator subscribed for your channel for the philosophical talks!
Very nice textbooks. I am reading the Robert. A. Adams, Calculus a Complete Course, for Engineering in Canada. Very rigorous and nice problems.
In what phase do you study Spivaks book in citytutoring and how much of the exercises do your students do?
I also like Apostol. Vol II is one of the best tie-ins of linear algebra to differential equations that I’ve seen. He also treats integration first, before differentiation, which is historically what was done before modern texts.
Yes! I really like that about Apostol! Integration should indeed be taught first!
@@citytutoring Christmas greetings! Just this morning this "math hobbyist" came across your channel, and two videos in I was hitting the subscribe button. I'm curious if you've had the opportunity to look at John Olmsted's two-vol calculus work. From 1968, it also takes the "integration first" approach. I haven't plunged into it yet, but it appears to place a premium on rigor. I'd love to hear your quick take on it, should you have already had it out for a test drive by chance. Thank you.
Spivak was the text in my first calculus course at my university.
it was mine too, it was very theoretical and proof based
What do you think of Europe's approach of not having Calculus and then Analysis but gouing straight to Analysis without having Calculus first? Also, what do you think of Soviet books on maths? I quite like the Fichtenholz course and that of Ilyn and Pozniak.
I very much liked the Soviet era books on maths! One of my favorite school level Algebra books was Israel Gelfand's Algebra. As for your question, I think it works well for Europeans, but would be very confusing for Americans who are not used to having to prove things.
By starting directly with analysis, students are introduced to rigorous mathematical thinking from the outset. This builds a solid theoretical foundation, which can be beneficial for students who pursue higher-level mathematics. The distinction between calculus (focused on computational techniques) and analysis (focused on rigor and proofs) is somewhat artificial. Combining them into one course can help students see the bigger picture and connect theory with applications early on, so I do like that approach, but think it would be (currently) unworkable in America due to the way our educational system is structured.
Spivak? lets go I use that too. It's not bad
Having a basic understanding of how something works is important, but you can't let yourself get bogged down in the details. An extreme example of this is the "Jordan Curve" theorem in complex analysis. Easy to state, easy to demonstrate, seems totally intuitive, but difficult in the extreme to prove. First time through the material, you give plausible arguments of why something is true or not. Once a good understanding of the material and how to use it, then a second time through with more scrutiny and so on.
The Jordan Curve Theorem should actually be a reminder that intuition is a fickle companion in mathematics. It’s like the appearance of simplicity in a mathematical result is directly proportional to the agony of the proof. In mathematics, we don’t just 'understand' something on a surface level - we dissect it until even the smallest detail no longer hides in the shadows. I can appreciate being satisfied by intuition, but it never completely convinces me to be happy enough, because even the simplest truths can demand Herculean efforts to dissect. The beauty of mathematics is that every second pass reveals more of its intricate layers. I don't approach it as 'getting bogged down' - it's about showing that you have the courage to follow those rabbit holes wherever they lead, no matter how convoluted they may seem. That in and of itself can be very hard, but to me, worth it.
@citytutoring I think we're on the same page. Mathematics is a little like learning a foreign language. Say you are learning French, and you are reading a short story in the language. You go through it and try to understand as much as you can. If you don't know something, you try and guess the meaning from the context. If you get hung up on letting each word or construction, you don't have a complete understanding of, stop you cold, you won't get anywhere. The same with learning mathematics.
@@OleJoeMathematics is a language after all, the language of the universe, and the representation of logic.
I started my real analysis journey with rudin's principles of mathematical analysis. I must admit, I dislike this book at an undergraduate level. However, having suffered through it, more or less, I find reading other math texts much easier in a sense. I just picked up spivaks calculus and the text is so inspired and rich, in some ways I wish I'd started with it.
Just took my Precalculus final! It went well, considering it was multiple choice and easier than a regular chapter test haha!
Glad to read that! How did you do on your Vectors test? Hopefully you are not using a Cengage text!
It went well, an 85% And the textbook I use is Precalculus with Trigonometry by Paul Forester, but it’s mainly on and off and not always straight from the textbook!
I would really love a review of spivak calculus. I clicked on the video because it being the thumbnail
I love Adams and Essex, it' so under used, at least from what i've seen. From 8th edition on, it has the best coverage of multivariable and vector calculus that you'll find in a non specialized book.
Yes, absolutely! I also like how they clearly distinguish definitions without much clutter. They have excellent explanations in their examples as well.
Well would it be okay for a student to be introduced to a concept through one of these textbooks that avoid proofs to make the subject easier and then delve into the concept further with more rigor?
I agree with some of your points but there is nothing wrong if one starts with Stewart and ends with Spivak. There is nothing wrong in doing "both" kinds of books, one that is heavy on exercises and one that is more on rigor and formalism.
Quite right -- the problem is that often, unless a student is truly invested in the subject, they will quickly start to feel very lost even with Stewart. Many colleges today are using "Cengage" along with Stewart, which has been a disaster for first year college students. Stewart often leaves incomplete proofs, and two or three examples are not enough to understand many of the exercises unless you have backup material.
@ Agreed.
1:20 yes as a student I can tell you first hand, the school systems are especially terrible... I'm now in college but, especially in high school, it was an absolute joke... To go anywhere you have to self study these days, school systems will actually prevent you from going ahead because it makes other people feel bad.
Indeed! Unfortunately, the school problem started decades ago, and it gets worse (At least at the public school level, but even charter schools, every year). Already when I was in high school, I had issues with the lack of seriousness in the classrooms, but now when I look back, even my worst classmates would now look like honors students compared to some of the individuals today, including, sadly, some instructors who took it upon themselves to pretend to be a "buddy" to teens rather than BE AN EXAMPLE to young people of hard work, and honor.
@@citytutoring Yes, the whole "buddy pal" thing is not only counterproductive but it's also totally strange. You hit the nail on the head with that, though, I can tell you as a recent high school graduate, I'm going to put things bluntly, seniors struggle to read basic texts out loud, I noticed. And my school district was rated super high lol...
What are your views on Tom Apostol Calculus, Mr. Cromwell?
What's your issue with diagrams?
Currently taking ap calclulus ab right now am I am surprised at how there is no epsilon delta proof concepts. There is not much creative thinking that goes into this course. There is no derivation of formulas or proofs. This sparks my interest for that dolciani book on modern intro analysis (1964)
Of course you are a pure mathematician. Not shocking. 😅
Hi professor, I wanted to know your opinion on a book named "what is mathematics?" By Courant, provided you read it obviously. As I actually started to notice the dilemma you stated here which is watering down mathematics and sciences in general. since I started my calculus on my own with spivak's and another not so sophisticated calculus book together, but in the end I choose spivak's for it's rigor. So I would be glad to know what your opinion is on it.
I very much like the original version, written in the 1940s, because the abstract content is higher than in the newer revisions that were published many years later. The title, I believe, is very ambitious. Courant does justice to the classical elegance of mathematics. If I were in the mood to be diplomatic in a hostile crowd, I might be tempted to use his book as a reference, since his tone is more accessible to the masses.
It's certainly a fine introduction for the curious, but more like a very good quality appetizer before the main course.
What I most admire is how Courant frames mathematics as both an art and a science. He’s unflinching in his demand for rigor but also conveys the profound beauty of the subject. His work reminds us that mathematics is not just about solving problems-it’s about understanding the universe, about truth itself.
@citytutoring thanks!
Hello, out of curiosity have you thought about making a video on proofs? For me personally at least it's never been explained what exactly makes something a rigorous proof and I think a lot of people who're interested in math could benefit from it. Thank you for the book recommendations they really mean a lot. Hope you have a great day.
Yes! I do plan on a video series on proofs! You are correct that proofs are something very much needed! Thank you for your feedback as well because that is immensely helpful to me.
What do you think of Calculus: A Rigorous First Course by Daniel J. Velleman?
Not a bad choice, but not as rigorous as Spivak's. However, it is adequate enough for students who need some guardrails to make the necessary transition, so I would not mind recommending it, or even using it in class when necessary.
I think it’s a good textbook.
Very glad to hear Prof Cromwell weighing in on this one, as I like Velleman's work. As Prof Cromwell says, Velleman puts training wheels on the rigor, but I've gleaned a lot of benefits therefrom. To me, Velleman's strong suit is instilling in the reader the habit of approaching every problem with a very careful, methodical, "what am I overlooking that might invalidate my proposed answer?", mindset. If you've seen his "How To Prove It" text, his calculus text has the approach you'd expect.
Do you think you can make a video about fourier transforms?
I plan to, at some point, yes...when I start a series on Analysis.
What book should i use for multivariable calculus after spivak? (Im reviewing by reading this one)
What about Morris Klines' calculus book, Calculus An Intuitive and Physical Approach?
I have not personally used the book, but I have seen it a few times and looked through it. Seems to be OK and detailed enough for self-study.
I really don't think there is a way from the likes of Stewart to Papa Rudin.
Michael Spivak suggests that the audience for Calculus is students who have already learned calculus .
That makes sense given that many students who are using the book have taken some form of Calculus either with AP (at the high school level) or in some other program and are ambitious.
Reformist "drivel!" about explains my "education" and I am in my late 60s.
Rigour comes after the basic fundamentals have already been established.
The “applications” you speak of at 7:00 can be important since it gives students an opportunity to assign value to what they’re learning and gain a reason to care about mathematics. Only giving colorful pictures of “applications” is insufficient however, there needs to be a balance of both. Most middle school to high school students will not care about mathematics simply for the sake of mathematics, in most cases there must be some tangible reward associated with understanding mathematics beyond simple intellectual merits (such as career applications) for students to engage with learning mathematics. This is less of an issue of philosophical “right or wrong” and more so a practical mechanism designed to hook the interest of young people who may be considering their future career pathways.
Caclusils
Most college calculus courses are introductory and meant to prepare students for the sciences, not to become mathematicians.
I am
You are? What?
"...therefore I think." --Rene's lesser-known brother, Converse Descartes.
No, morality is not declining: th-cam.com/video/3sAxMG88xf8/w-d-xo.html
I don't equate "less violence" with "morality is not declining" necessarily. I am familiar with the argument that there is "less violence" in the world today, but that does not mean moral principles are not in decline. And, to be clear, when I say "moral principles", I specifically mean Christian moral principles.
Sure, fewer people are being burned at the stake by Roman Catholic tyrants or slaughtered in open battlefields, but let’s not mistake fewer swords for better souls. The real story isn’t about declining violence; it’s about shifting forms of immorality. Today, people destroy reputations instead of villages, exploit citizens through a thoroughly unholy alliance between government and corporate, and weaponize misinformation instead of loading muskets.
Violence may have declined, but deceit, apathy, and exploitation have found new, bloodless arenas to thrive. If morality were truly improving, we’d see more than just fewer corpses-we’d see a society less ruled by perversion of children, dishonesty, profanity, apathy, and the state destroying the lives of people who value freedom and independence. Measuring morality by body count? That’s setting the bar pretty low, don’t you think?