Book for self-learning calculus and learning in a foreign language?

[user079622]

1. Does it make sense to learn math in a foreign language(English) from book because I will take some engineering courses on English or better stay at mother language?

2. What book do you recommend for self learning calculus, maybe better engineering oriented?

 

[jedishrfu]

I would recommend learning Calculus using a book written in your native language.

You don't need the distraction of struggling to understand a foreign language while learning a hard subject.

However, while you're doing that, you could also read the English version of your book on the side mapping the terminology from your native language to English. Having both versions of the same book could help for future reference as well.

---

One book, I like is Keisler's book on Calculus because it uses hyperreals in its early development so that you learn about limits later than earlier. However, its considered non-standard because of the hyperreal basis of the book.

https://people.math.wisc.edu/~hkeisler/calc.html

Spivak is suitable for self-study, but the gold standard/most popular is the Stewart book, although there may be more recent, better-quality books.

Here is a list of the top five:

1. “Calculus” by James Stewart

• Why it’s popular: Clear explanations, a large number of examples and exercises, and accessible writing. It’s especially popular in engineering and science programs.
• Best for: Students who need practical calculus skills and application-focused learning.

2. “Calculus” by Michael Spivak

• Why it’s revered: Provides a rigorous, proof-based introduction to calculus, blending analysis and formalism with a focus on conceptual understanding.
• Best for: Students interested in mathematics or preparing for advanced study in pure math.
• Style: Challenging, with an emphasis on proofs and theory.

3. “Calculus” by Tom Apostol

• Why it’s exceptional: Offers a formal, axiomatic approach, introducing integration before differentiation, which provides a unique and deeper understanding of the material.
• Best for: Highly motivated students, particularly those interested in a theoretical foundation and connections to linear algebra.
• Style: Rigorous and structured.

4. “Calculus: Early Transcendentals” by William Briggs and Lyle Cochran

• Why it’s effective: Focuses on clear explanations and real-world applications, balancing rigor and accessibility.
• Best for STEM students looking for a practical yet comprehensive introduction.

5. “Introduction to Calculus and Analysis” by Richard Courant and Fritz John

• Why it’s historic: Written by a legendary mathematician, it provides a deep, intuitive, and rigorous approach, with a focus on physical applications and mathematical thinking.
• Best for: Students with strong mathematical maturity seeking a comprehensive and challenging text.

Some honorable Mentions:

• “Calculus Made Easy” by Silvanus P. Thompson and Martin Gardner: Great for beginners or those struggling with intuition.

• “Advanced Calculus” by Patrick M. Fitzpatrick: A bridge between undergraduate and graduate studies.

• “Elementary Calculus: An Infinitesimal Approach” by H. Jerome Keisler: Introduces calculus using non-standard analysis.

 

[user079622]

@jedishrfu
So Stewart is no.1
What do you think about book Thomas Calculus Early Transcendentals?

 

[fresh_42]

@jedishrfu
So Stewart is no.1
What do you think about book Thomas Calculus Early Transcendentals?

https://www.physicsforums.com/threa...-old-physics-enthusiast.1044657/#post-6789594

https://www.physicsforums.com/threa...ls-textbooks-for-mst124.1017424/#post-6660463

https://math.stackexchange.com/ques...arly-transcendentals-and-late-transcendentals

 

[user079622]

@jedishrfu
@fresh_42 thanks for explantion

What Calculus of James Stewart you suggest, there are more versions?

Do you agree with this text ?
This opinion I find here: https://www.quora.com/What-are-the-best-calculus-books

Quote from text:
"The problem with books like Thomas’ Calculus or Stewart Calculus is that you won’t get a thorough understanding of the inner mechanics of calculus. As long as you don’t have a good prof or teacher, I would stay away from these books. If you want to understand what I mean, take a look at some arbitrary sections in these books. You’ll see a short paragraph, which serves as an intro, then some boxes with formulas, then a few workout examples and then a bunch of exercises. This means, you will only learn HOW to you the formulas instead of understanding the WHY!

My advice is, visit YouTube, search for Michael Van Biezen, learn the techniques of Calculus 1–3 (ca. 17 hours), and then, to understand the inner mechanics of Calculus, read Tom Apostol. Biezen will serve as a shortcut for learning the techniques and Apostol will teach you the WHY.

Alternatively you can search for Prof.Leonard on YouTube and watch his Calculus 1–3 lectures (ca 168 hours). He works through the books like Stewart Calculus but tries to teach you the sections in detail. Nevertheless, I would prefer the first way Biezen -> Apostol.

To answer your question,

• Gilbert Strang - Calculus (very good, but in my opinion to conversational. You can find it for free on the website of MIT)
• Tom Apostol - Calculus (very very good, but you need to put in serious effort)
• Michael Spivak (didn’t read it, but many people say, it’s quite harder than Apostol, but still one of the best books to learn Calculus, although only Single Variable Calculus)
• Serge Lang - First Course in Calculus (makes fun to read it, built more on intuition)
• Thomas’ Calculus (short on explanations and too dry)
• Stewart Calculus (same as Thomas’ with the exception, that he has more real world examples)"

 

[fresh_42]

I am no fan of YouTube videos except for music, movies, or other entertainment. The reason for this is already part of my answer. It's entertaining and the authors sell the imagination of having understood the matter. But that's exactly the problem. It is an imagination: lean back and enjoy is not how studying works. You have to go through the matter working it out. Consumption is not the way to do this.

I share the opinion that Stewart's book is very example-heavy and not focused on the underlying mathematical principles. But that might not be necessarily the worst approach for a future engineer.

I was surprised that Serge Lang's book was associated with fun. I like the Bourbaki approach to mathematics and Lang was part of the Bourbaki group. And you can't go wrong with Lang, since he was a very competent mathematician.

If you want to study calculus at the mathematics or physics level then I would choose a classical book like Spivak's Calculus. However, I want to emphasize that I fully agree with ...

I would recommend learning Calculus using a book written in your native language.

It all depends a bit on what you already know, what your goals are, and last but not least what suits you best in the sense of how you learn, memorize, and understand best. Recommendations on the internet usually cannot consider these very important individual aspects.

 

[fresh_42]

Here are some tricks on how to use language issues to your advantage:
https://www.physicsforums.com/threads/how-to-use-the-w-in-www.1062388/

I would search for <"calculus 1" in your native language + pdf> to find lecture notes on university servers in your country.

 

[user079622]

I am no fan of YouTube videos except for music, movies, or other entertainment. The reason for this is already part of my answer. It's entertaining and the authors sell the imagination of having understood the matter.

But here is example where proffesor recorded his class at calculus, so there is not difference if I am in class or in my room. Problem is, bigginer cant understand everything in new topic from book without someone explain him.
I professor is good, he will point out the most important things.

I want to emphasize that I fully agree with ...
"I would recommend learning Calculus using a book written in your native language."

Yes I agree that learning in native language is always better.
I just ask because all courses are in english, this course need knoweldge in vectors, algebra , calculus etc
We have english language in all software, because all explantions at youtube are in english...

I share the opinion that Stewart's book is very example-heavy and not focused on the underlying mathematical principles. But that might not be necessarily the worst approach for a future engineer.

She study math and physics, said in video that use J.Stewart at university.
I took a quick look of this book, to me seems more like tasks-book (compare to I had). (Even this is OK for enigneer)
But why they use this book at math university?

4:25

 

[fresh_42]

But here is example where proffesor recorded his class at calculus, so there is not difference if I am in class or in my room.

That's part of the illusion. It is a difference whether you observe the people next to you in a café or watching a movie scene of people sitting in a café. For instance, I like to watch Feynman lectures on YouTube a lot. But I do this for entertainment and for getting inspiration. I would never claim that I have studied anything while watching those videos because I haven't!

But why they use this book at math university?

Not enough information.

What are the typical students of this university? E.g. Americans typically leave high school having learned far less than Germans leaving school here. And here it even depends on regions! Which students are addressed? Only mathematicians, or students of all kinds of faculties as a basic course for all of them? What are the goals of the course?

I wouldn't use Stewart if I held calculus 1. That doesn't mean someone else would.

 

[user079622]

That's part of the illusion. It is a difference whether you observe the people next to you in a café or watching a movie scene of people sitting in a café. For instance, I like to watch Feynman lectures on YouTube a lot. But I do this for entertainment and for getting inspiration. I would never claim that I have studied anything while watching those videos because I haven't!


Not enough information.

What are the typical students of this university? E.g. Americans typically leave high school having learned far less than Germans leaving school here. And here it even depends on regions! Which students are addressed? Only mathematicians, or students of all kinds of faculties as a basic course for all of them? What are the goals of the course?

I wouldn't use Stewart if I held calculus 1. That doesn't mean someone else would.

I dont know, she is from Australia.

I learned from videos of Professor Leonard lots of things, because it didnt write in my book or becuase I didnt conclude what does it really mean.

I just tried to read a little text from Spivak pdf, yes it is much better in may native language.
Sometimes I read a sentence and have to translate what I just read in my head. Only then can I think about what it exactly means. Too slow..

 

[jedishrfu]

I like the videos from mathispower4u.com

They are short, concise and focus one problem at a time.

The author doesnt get hung with fancy graphics to dazzle the eye.

There are roughly 5000 videos covering a range of math subjects including Calculus 1,2,3 linear algebra and differential equations.

You can follow along with video or stop it and attempt to solve a step before moving on.

 

[user079622]

I like the videos from mathispower4u.com

Amazing how many videos that site has.

Better then Khan?

 

[Muu9]

What is your native language?

 

[jedishrfu]

There is no best book. There are many great books, all with varying perspectives on how to teach Calculus based on the student's focus.

 

[user079622]

I would recommend learning Calculus using a book written in your native language.

You don't need the distraction of struggling to understand a foreign language while learning a hard subject.

However, while you're doing that, you could also read the English version of your book on the side mapping the terminology from your native language to English. Having both versions of the same book could help for future reference as well.

---

One book, I like is Keisler's book on Calculus because it uses hyperreals in its early development so that you learn about limits later than earlier. However, its considered non-standard because of the hyperreal basis of the book.

https://people.math.wisc.edu/~hkeisler/calc.html

Spivak is suitable for self-study, but the gold standard/most popular is the Stewart book, although there may be more recent, better-quality books.

Here is a list of the top five:

Some honorable Mentions:

From Europe perspective Stewart is workbook not book. But now I see that exercises dont even have answers, that is unacceptable. Now I understand what quote from Quora want to say.

Why Apostol introducing integration before differentiation, what is logic?
Isn't that like learning to divide before adding?

 

[Muu9]

From Europe perspective Stewart is workbook not book. But now I see that exercises dont even have answers, that is unacceptable. Now I understand what quote from Quora want to say.

Why Apostol introducing integration before differentiation, what is logic?
Isn't that like learning to divide before adding?

Integration is continuous addition, differentiation is infinitesimal division.

 

[fresh_42]

Integration is continuous addition, differentiation is infinitesimal division.

That's a dangerous point of view. It invites to misconceptions.

 

[gmax137]

Why Apostol introducing integration before differentiation, what is logic?

Integration and differentiation are two faces of one idea.
See the wiki:
Fundamental theorem of calculus

 

[Muu9]

Integration and differentiation are two faces of one idea.
See the wiki:
Fundamental theorem of calculus

This isn't really an argument for deliberately choosing to introduce integration before differentiation

 

[user079622]

This isn't really an argument for deliberately choosing to introduce integration before differentiation

I think this is only book with that order?

 

[fresh_42]

Why Apostol introducing integration before differentiation, what is logic?

That's impossible to tell if it isn't said in the book. Maybe it is because it is easier to visualize Riemann-Darboux integrals as "area below the curve" whereas differentiation leads to tangents that could be harder to comprehend in the sense that the uniqueness of the limit of a sequence of secants isn't obvious.

Isn't that like learning to divide before adding?

I would strongly recommend not to compare differential calculus with basic operations. Integration is not an addition; it is an oriented volume closer to multiplication than to addition. And differentiation is far from being a division. It is a linear approximation: find a function $D(x)$ such that
$$
f(x+v)=f(x)+D(x)\cdot v +r(v)
$$
Where do you see a division here?

Another aspect why such a comparison is problematic, is the simple fact that differentiation is easy and integration difficult, whereas addition is easy and division is difficult - the other way around.

 

[Muu9]

Where do you see a division here?

Solving for D(x) requires dividing by v

 

[fresh_42]

Solving for D(x) requires dividing by v

You cannot divide by $v$ since $v$ is a vector! You can only renormalize its length to one. The fact that a vector and a scalar look the same in one dimension is a coincidence, and as such shouldn't be taken to draw conclusions. That's exactly why I oppose the comparison to division. It is by chance, not by cause!

 

[user079622]

That's impossible to tell if it isn't said in the book. Maybe it is because it is easier to visualize Riemann-Darboux integrals as "area below the curve" whereas differentiation leads to tangents that could be harder to comprehend in the sense that the uniqueness of the limit of a sequence of secants isn't obvious.

Integration before derivation is very unlikely, in math and physics classes/books. I dont like that order.
Order must make be logic, you first learn derivation, then antiderivation.

 

[fresh_42]

Integration before derivation is very unlikely, in math and physics classes/books. I dont like that order.
Order must make be logic, you first learn derivation, then antiderivation.

I agree, but it can be done. Riemann-Darboux integration is more intuitive than a limit.

 

[semperfidelis]

Stewart or Thomas are almost like Schaum's, just problem practice. The other books mentioned (Apostol, Spivak, etc) serve for real insight/conceptual learning. So when a post above mentioned Stewart as a gold standard, it depends in which type of college. I'd assume that top colleges use insight textbooks, not workbooks.

 

[Muu9]

Stewart or Thomas are almost like Schaum's, just problem practice. The other books mentioned (Apostol, Spivak, etc) serve for real insight/conceptual learning. So when a post above mentioned Stewart as a gold standard, it depends in which type of college. I'd assume that top colleges use insight textbooks, not workbooks.

The book used by Harvard is closer to Stewart than Spivak: https://people.math.harvard.edu/~knill/teaching/math1a2024/handouts/1a_2024.pdf

Dartmouth uses Stewart: https://math.dartmouth.edu/undergraduate/first-year-students/placement-info/syllabi/m3-syllabus.php

Cornell uses https://activecalculus.org/, which is mainly problem practice

UChicago uses Salas and Hille; again, more like Stewart: https://vipulnaik.com/math-152/

Etc, etc.

 

[semperfidelis]

What about more science focused schools like Caltech or MIT?

 

[fresh_42]

What about more science focused schools like Caltech or MIT?

I think that "self-learning" shouldn't follow any specific university. The basics are the same everywhere, and dozens of free lecture notes are available. Choosing one that suits one's individual fondness rather than sticking to a dogmatic call for a certain university is fairly easy. Calculus is the same everywhere. In Edinburgh, Toronto, Kaiserslautern, Boston, or Bournemouth.

 

[Muu9]

What about more science focused schools like Caltech or MIT?

https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/mitres_18_001_f17_full_book.pdf

Caltech used (maybe still uses) Apostol I'm the first quarter. Interestingly, it is the only US college to have calculus in high school as a hard requirement for admissions. More interestingly, for the second and third quarters students are given the option to switch into a more applied track, which many take.