What Is the Best Calculus Book for Self-Learners?

[Heisenberg7]

Hello,

I would like to start off my saying how much Calculus I have done so far. I am familiar with the idea of limits, derivatives and integrals (though I do have some holes in my knowledge). So far, I have only done Calculus I. I was introduced to some ideas of Calculus II, but those were mostly from YouTube videos.

For the past week and a half, I've been studying proofs (book of proof by Richard Hammack). The thing is, I am not sure if I should be studying proofs. At this moment, I can say that I am terrible at writing proofs. I do know some techniques (e.g. proof by contradiction) and I understand them really well. But, I am terrible at producing them myself.

Another problem I have is finding the right Calculus book. I tried using Spivak's Calculus book. I understand it pretty well, but there is a huge problem. I can only do about 1/3 of the exercises from each section. Some problems require you to know how to prove things. That's the reason why I set out to learn how to do it. I tried using other Calculus books, but they don't even come close to Spivak's book (when it comes to understanding) and I feel like I'm just wasting my time on them. I would like to master most of the Calculus I and II within a year.

What should I do? Also, I am self-taught.

Thanks in advance

 

[Frabjous]

You might check out the books by Apostal and Courant. They are at a similar level.

This thread has some other suggestions.
https://www.physicsforums.com/threads/calculus-book-between-stewart-spivak-levels.1049989/

You should search PF for threads with Spivak or Apostal. You should find things of interest.

Is your goal advanced math or science/engineering? If it is science/engineering, Spivak level is more than you need.

 

[Heisenberg7]

Is your goal advanced math or science/engineering? If it is science/engineering, Spivak level is more than you need.

It's science/engineering. I figured that Spivak might be more than I need.

Should I finish the book of proof and then do Calculus?

 

[Frabjous]

It cannot hurt. If you still cannot handle Spivak, you will then need to move to something simpler.

You might have to do something simpler regardless, in order to internalize all of the applied techniques.

 

[WWGD]

As always, I suggest that, if possible, you drop by a library containing Calculus books, browse through them for a few hours , and see which one feels right for you.

 

[marcusl]

Try Thomas, Calculus and Analytic Geometry. It’s been used by generations of college students. You don’t need a new copy, a cheap used copy of the fourth edition (or 5th or 6th) will do fine.

 

[WWGD]

There are too, Paul's Notes, freely available,
https://tutorial.math.lamar.edu/

and great YouTube lectures by Trefor Bazzett(Sp?).

 

[nordoN]

You might check out the books by Apostal and Courant. They are at a similar level.

This thread has some other suggestions.
https://www.physicsforums.com/threads/calculus-book-between-stewart-spivak-levels.1049989/ 22bet

You should search PF for threads with Spivak or Apostal. You should find things of interest.

Is your goal advanced math or science/engineering? If it is science/engineering, Spivak level is more than you need.

I like Spivak's and Apostal's books as they have a practical bent.

 

[Muu9]

I would honestly recommend a standard text, you can find reviews here: https://forums.welltrainedmind.com/topic/604324-my-review-of-various-textbooks-for-calculus/

You can often find used copies very cheap using bookfinder.com

 

[mathwonk]

Since you seem to prefer Spivak for its explanations and you can do 1/3 of the problems, you might just supplement your work by consulting his answer book. Here is the combined answer book for editions 3 and 4.
https://www.amazon.com/Combined-Answer-Calculus-Fourth-Editions/dp/0914098926/?tag=pfamazon01-20

Since Spivak explains mainly the theoretical parts of calculus, I also suggest supplementing it with another book that includes applications. Courant does that and is almost as mathematically rigorous as Spivak.

here are some cheap used copies:
https://www.abebooks.com/servlet/Se...-_-Results&ref_=search_f_hp&sts=t&tn=calculus

here is the revised version by (Courant and) Fritz John, also excellent.
https://www.amazon.com/Introduction...s-Mathematics/dp/354065058X?tag=pfamazon01-20

here is an international edition of Apostol, apparently legally available in the US but not in Canada, also outstanding:
https://www.abebooks.com/servlet/BookDetailsPL?bi=31445466021&cm_sp=SEARCHREC-_-WIDGET-L-_-BDP-R&searchurl=ds=30&kn=apostol&rollup=on&sortby=17

More applied, but still recommended, is this acceptably early edition of Thomas' Calculus, actually written by Thomas! (Having taught from them, I suggest avoiding any editions co-authored by Hass and/or Weir.)
https://www.abebooks.com/servlet/Bo...=60&tn=Calculus&cm_sp=snippet-_-srp2-_-title9

This one by Thomas and Finney should be ok, and is under $4!
https://www.abebooks.com/servlet/BookDetailsPL?bi=31905470954&searchurl=an=George+b.+Thomas&ds=30&rollup=on&sortby=17&tn=Calculus&cm_sp=snippet-_-srp0-_-title12

 

[berkeman]

Welcome to PF.

I like Spivak's and Apostal's books as they have a practical bent.

Can you give a specific example from the book? I would be interested in what you consider a practical problem. Thanks.

 

[mathwonk]

yes that remark piqued my interest as well. there are applications in apostol in chapters 2,4,8,14, but the word "volume" does not even occur in the index of spivak, and "area" only occurs a couple times, mainly to motivate integrals. the words mass and velocity occur very minimally, without any illustrations or calculations.

dear nordoN, are you a bot? or can you explain how spivak could possibly be considered "practical"?

 

[berkeman]

dear nordoN, are you a bot? or can you explain how spivak could possibly be considered "practical"?

Exactly.

 

[MidgetDwarf]

If you want something easier than Spivak,Courant,Apostl.


Then it would be Moise: Calculus.

The issue with Moise is that its been out of a print for a few decades now, and copies are scarce...

Maybe a google search will find the book for yoi.

 

[mathwonk]

https://www.amazon.com/dp/0201047950/?tag=pfamazon01-20

but compared to what we have found, I do not recommend spending $60 on this book. If you are curious to see if it appeals to you, you can likely find it in a library and judge for yourself.

 

[Frimus]

Here you can find free gems (no strings attached):
https://classicalrealanalysis.info/Free-Downloads.php

 

[vela]

I tried using other Calculus books, but they don't even come close to Spivak's book (when it comes to understanding) and I feel like I'm just wasting my time on them. I would like to master most of the Calculus I and II within a year.

Can you tell us how do you decide when you've understood a topic to a sufficient degree?

Learning isn't the linear process many make it out to be. In physics, for example, you don't learn classical mechanics in all its glory in one go (in the US). You see it first in intro physics, mostly learning how to solve basic problems and get numerical answers. You study it again in an upper-division course, where you're exposed to it at a more sophisticated level, and then if you go to graduate school, you'll revisit the subject again at an even deeper level.

With calculus, your time may be better spent at this level at focusing on calculations and understanding proofs, rather than learning how to write rigorous proofs. (Writing proofs is a skill many students find difficult to learn. As your critical thinking skills get better, you'll probably find it easier to write rigorous proofs.) Later, once you're familiar with the lay of the land, you can go back and learn how to prove these results rigorously.

 

[Vanadium 50]

I'm not sure this strategy of yours is working. You seem to be studying physics and calculus on your own, at the same time, and trying to go quickly. You will likely find that if you go any faster, you'll never finish. Get your foundations in order and things will go smoother.