# How to Self-Study Calculus - Comments

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• micromass

#### micromass

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How to Self-Study Calculus

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bhobba, bentleyghioda, KSG4592 and 2 others

Nice one micromass! I've always thought about what this kind of list should constitute, and you've covered it really well. Before this, I was forced to say "you need Calc I, II, III and DEs to understand physics well" to my friends, but had a really hard time explaining the contents of each in detail. Well, I have a great reference now

Thanks a lot PWiz, I appreciate it. If you think I've missed something, please do tell!

Thanks a lot PWiz, I appreciate it. If you think I've missed something, please do tell!
IMHO, "parametric equations" and "calculus in different coordinate systems" (or something along those lines) should be included in the post somewhere under the Multivariable section, but other than that, I think your post pretty much covers all the bases.

Yes, very good! I will edit this in.

I would like to share a recommendation: G.M. Fichtenholz "Differential and Integral Calculus". Fairly unpopular outside of the 'post-Soviet' countries, but it is among my personal favourites. A bit on the lengthy side, but it keeps a very approachable and 'eager to explain' tone just as easily when talking about basic differentiation and application of multi-variable functional series and transforms. Book genuinely 'feels' like a transcript from a very patient tutor. Plus it makes it a point to show worked-out examples to almost every single concept.

It is also among the most complete resources when it comes for computational techniques, so if not for any other point it is still worth at least as a reference on solving problems.

I will be learning calculus for the first time shortly and it's nice to have a guide like this. Your posts are tremendously helpful for beginners like me, it is much appreciated!

bhobba and Greg Bernhardt
I'm afraid you posted a wrong link for the "Calculus in 3D: Geometry, vectors and multivariate calculus by Nitecki".
This works better: http://www.tufts.edu/~znitecki/Hardcore2.pdf

micromass
You seem to be presenting this as a "one true way". For instance, you say differentiation is a prerequisite for learning integration. I'll note that Apostol does it the other way around, likely because historically that's the way it happened. Do you think that self-teaching from Apostol is a bad idea?

I'm just thinking you might want to make the tone a bit more "here's one way to do it" than it is now. In any case, kudos on recommending free texts! That's certainly one thing Apostol's Calculus does not have going for it, regardless of how good it might be.

You seem to be presenting this as a "one true way".

I don't think I have said or implied anything remotely like that.

You seem to be presenting this as a "one true way".

I don't think I have said or implied anything remotely like that.
Lest I be misunderstood in offering criticism, let me say thank you for doing this. It's a meritorious effort and will be helpful to many, I'm sure.

My impression regarding it being presented as the "one true way" came from these statements:
The best calculus book is undoubt[ed]ly... (highly controversial)
So it is very beneficial to learn the nonstandard approach. (controversial at best)

But I agree those are not representative of the whole piece. However, the impression I get is that you think these textbook suggestions are right for everybody. I've found that people have different styles and need different things. Some love examples, some hate them. Some need rigor and others prefer intuition. Some like exercises aplenty, and others prefer a few well-chosen problems. Some want an answer key and others find it too tempting and prefer it doesn't exist. Some want their mathematics pure and others find it dry as dust if there isn't real world motivation.

It would help, I think, if you indicate who your recommendations are for. If you really think they'll work for everybody, I'm suspicious.

The texts are just his recommendations. And it makes much more sense to introduce differentiation before integration.

bhobba and PWiz
Hmm interesting, of all those topics (it took me 2 semesters to get over them) the courses i took on the matter never talked about multi variable Taylor series, Laplace transform, or system of ODEs :c, maybe i should try to learn those on my own.
Also, about "vector calculus" section, does that mean Green's, Gauss' and Stokes' theorem?
Very good, organized, and easy to read.
Cheers :D

Hmm interesting, of all those topics (it took me 2 semesters to get over them) the courses i took on the matter never talked about multi variable Taylor series, Laplace transform, or system of ODEs :c, maybe i should try to learn those on my own.
Also, about "vector calculus" section, does that mean Green's, Gauss' and Stokes' theorem?
Very good, organized, and easy to read.
Cheers :D

Yes, vector calculus is stuff like Stokes' theorem.

Of course it is very likely that your courses did not cover everything of this. I don't think it is really absolutely necessary to go back and learn them on your own (unless you enjoy learning this stuff of course, in which case: go ahead). If you ever meet one of those topics later, you can still go back and learn them.

I just ordered the book of Mary Boas. How does that compare with these books?

bhobba
Boas is a math methods for physics and engineering. It has less emphasis on theory, and goes over different subjects such as LA, DE's, vector calc, basically everything an undergrad physics major will need. If you're a physics major It will benefit you tremendously to work through it.

bhobba
I just ordered the book of Mary Boas. How does that compare with these books?

It doesn't compare at all with these books. They are very different. First of all, Boas does not cover single variable calculus. It starts with series and multivariable calculus. So it assumes you know integrals and derivatives already.
Second and most important, Boas is for physicists who don't really care much about the underlying math. So if you want to know the math in detail, then Boas is not good. If you simply wish to use it as a tool, then Boas is truly an excellent resource.

Well, I'll just have to buy another book then...

Well, I'll just have to buy another book then...

What is your goal? What kind of book do you want?

What is your goal? What kind of book do you want?

I really want to understand the math behind quantum mechanics...

Send me a PM, I might be able to help :)

Hi,

What is your opinion on Courant's introduction books on calculus ?

Thanks

Nice to see Keisler's "An infinitesimal approach to calculus" in the list. Great post.

bhobba and micromass
Hi,

What is your opinion on Courant's introduction books on calculus ?

Thanks

It's a very nice book. But don't use it as first calculus book, since it's too difficult for that. It is very suitable as a second course though, if you enjoy the book.

bhobba and Physicaa
It's a very nice book. But don't use it as first calculus book, since it's too difficult for that. It is very suitable as a second course though, if you enjoy the book.

Can I skip first two chapters of third book if I followed the first book ?

Dear micromass,

Have you read Apostol's calculus books? I want to know how they compare to the two Nitecki books you suggested in your guide? And also how Freidberg's Linear Algebra compares to Shilov's book on the same topic.

I wanted to go through calculus and then Linear Algebra following either of two paths:
a) Keisler's Infinitesmal approach>>>Nitecki Deconstructing Calculus>>>Nitecki Calculus in 3D>>>Freidberg's Linear Algebra

OR

b) Simmon's Calculus with analytic geometry>>>Apostol Vol 1>>>>Apostol Vol 2>>>>Shilov's Linear Algebra

Apostol's calculus books are fantastic for a first course on analysis, i.e. for a SECOND phase on calculus in the standard pedagogical sequence for most people who want to study mathematics formally. They are too dense to be useful for a first course on single or multi-variable calculus.

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My recommendation after Elementary Calculus is either Boaz or Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard:
https://matrixeditions.com/5thUnifiedApproach.html

Personally, I would study both; Boaz then Hubbard. They each have different purposes - Boaz covers the math you need for the physical sciences, but not rigorously. Hubbard covers Multivariable Calculus and Linear Algebra with excellent rigour. I personally always like to go from a less formal approach to one that is rigorous.

Thanks
Bill

vanhees71