What is a good calculus book for reviewing Calc 1/2 and learning Calc 3?

[subsonicman]

Hi, I took Calculus BC in junior year of high school, and a multivariable calculus class in senior year that covered some partial derivatives and integration but didn't go into vector calculus. I've decided to review the subject myself over winter break (and, if need be, a little into next semester). Do you guys know of any good book(s) to get a really strong Calc 1/2 basis and is also good at teaching Calc 3? I am very strong in math so a book of any difficulty level is fine as long as it teaches the subject clearly.
I've heard of Spivak's for Calc 1/2. Is that a good choice?

Thanks in advance.

 

[chiro]

Hey subsonicman.

If you are looking at Calc 3 I recommend this:

https://www.amazon.com/dp/0805390219/?tag=pfamazon01-20

 

[MarneMath]

Spivak for calc 1 material and then his manifold book as Chiro recommended. I also recommend Apostol, not as fun to read as Spivak, but in his two volumes you'll get everything you need.

 

[Vargo]

My humble vote is against either of Spivak's books. I own his "Calculus" because at one point it appealed to me, but I have since decided that his book lacks a viable target audience. Perhaps I am wrong, but it is too hard for beginners and too lightweight (figuratively!) for advanced calculus. The OP might be in the happy middle ground who will actually benefit from the book, but I would recommend something with a more varied set of exercises instead. Spivak's exercises seem to me to be very focused on the art of proofs and preparation for heavier analysis.

Please forgive me if I think that Spivak's Calculus on Manifolds is insane for someone who has just completed BC Calculus.

Apostol's books are very comprehensive and have lots of great exercises. Also (unlike the "purist" approach of Spivak), he is not above the use of geometry to define things like the sine function, putting him more squarely within traditional conventions of introductory calculus. Yes, Apostol is also very dry and the text can be quite be quite boring, so there is that...

"Div Grad Curl and all that" is a good, short book that is exclusively focused on vector calculus.

 

[subsonicman]

The spivak manifolds book mentions a term of linear algebra but I haven't taken the class yet. Do I need a term of linear algebra to read the book?

But thanks for everyone so far, I'll look into Apostol and Spivak.

 

[subsonicman]

I've heard Courant volume 2 is a good multivariable calculus book, how is that?

Also, I should've mentioned in my original post, but I'm looking for a good text to learn beginning differential equations. I have no idea where to start (ODEs?) so any help would be appreciated.

 

[MarneMath]

Wellllll! I humbly disagree with Vargo. I don't believe BC Calculus prepares you at all for Spivak Calculus. I took an honor sequence calculus course that used Spivak at college, and naturally nearly everyone took BC Calculus, but nearly everyone struggled, except for some really brilliant people.

I'll readily admit that some of Spivak is not so great, ie the chapter on "Three Hard Theorems" is a bit hard to read and lacks motivation (due to the fact it ends up being a lot of delta pushing instead of using the words 'compactness.') Nevertheless, I think it eases the job from "This is Math we plug in numbers and that's it" to "This is Math, how does it work?"

I enjoyed Courant and reading it was a joy. It's a great multivariable book, but I suggest starting with his first volume in order to fill some expected gaps.

As for diffy q, a lot of people recommend Arnold, and I do too, but not until you have more mathematical maturiy. A lot of intro diffy q books are simply plug and chug, so if you're ok with that, then I tend to recommend boyce.

 

[Vargo]

I recommend Tenenbaum's Ordinary differential equations with reservations.

The good and the bad:
It is a Dover book, so you can get it for something like $15. At around 800 pages with not too many pictures, it is an encyclopedic mega-tome. It was written for the usual Intro to differential equations crowd, but in 1963, which in this case, is a good thing. You don't have to know linear algebra or advanced calculus to read it. It has a great many exercises, and it shows a wide variety of interesting applications that seem real, not contrived. It does have a hefty chapter on numerical methods, but, because it was written in 1963, it does not mention computers. It is probably a lot more information than you are looking for, but it has a lot of gems that can come in handy later (though you might forget them by then because no one remembers everything they read out of an encyclopedia). Anyway it is something to consider.

I agree with MarneMath that a book like Boyce and DiPrima gives you what you need to know at the plug and chug level. You could also try something like Paul's Online Notes:
http://tutorial.math.lamar.edu/Classes/DE/DE.aspx

 

[subsonicman]

So Marnemath, you're saying if I get Courant volume 2 to study multivariable then I should get Courant volume 1 rather than spivak to review one variable calculus?

And I'm unsure how far I want to go into differential equations. I want to have enough background to solve any physics problem involving them, but since I've barely even looked at differential equations before, save for very simple ones, I don't know how interesting the topic would be for me. Is it required to have a solid grasp of differential equations to study further in other math classes like analysis and topology?

 

[WannabeNewton]

So Marnemath, you're saying if I get Courant volume 2 to study multivariable then I should get Courant volume 1 rather than spivak to review one variable calculus?

Make sure you know what you are getting into before you spend money and time. The material you learned in AP Calculus BC will be a joke beyond joke compared to Courant. Courant is way more advanced. Spivak at least eases you into proofs, to some extent, if you have never seen them before - Courant does not. Most if not all Calc BC type problems can be solved with one's eyes closed but it doesn't work that way, in general, for problems in Courant and Spivak; there is a HUGE gap between Courant and AP Calculus BC. The same goes for calc 3 and Courant's multivariable. You can certainly go for it, and I wish you the best of luck and definitely hope you are successful in using the book(s) but before you spend the money make sure it is not too daunting a task. As for topology and analysis, you certainly don't need DEs to start studying them.

 

[subsonicman]

I've had a lot of proof experience, mostly outside of calculus but also some in calculus. As long as the book doesn't require subjects I haven't learned (like linear algebra) I should be fine.

 

[WannabeNewton]

I've had a lot of proof experience, mostly outside of calculus but also some in calculus. As long as the book doesn't require subjects I haven't learned (like linear algebra) I should be fine.

You will definitely need LA for calculus on manifolds (Spivak) but not for the single variable book.

 

[MarneMath]

So Marnemath, you're saying if I get Courant volume 2 to study multivariable then I should get Courant volume 1 rather than spivak to review one variable calculus?

And I'm unsure how far I want to go into differential equations. I want to have enough background to solve any physics problem involving them, but since I've barely even looked at differential equations before, save for very simple ones, I don't know how interesting the topic would be for me. Is it required to have a solid grasp of differential equations to study further in other math classes like analysis and topology?

No. I'm saying if you picked Apostol, Spivak, and or Courant, you'll have be doing fine. Apostol, has an interesting approach and a geometric approach that is valued by some people. Spivak is easy to read, and helps you out with proofs (ie by breaking them down into smaller easier problems or giving big hints.) Courant is fun to read and very similar to Spivak (to the point I think Spivak read Courant and then put it in his own words.)

I tend NOT to recommend Courant for self studying just because it has a lot less exercises and even though a lot of problems are the same as Spivak, the lack of hints can make them inaccessible to a student not prepared. Still, if you can find a cheap copy, it's a a good read by a great mathematician.

As for multivariable calculus, I tend to recommend to have linear algebra before taking it. Although it's extremely common to learn the subject without LA, I've found that if you know it already, then it's easier to generalize and see why certain things are done in a multivariable book. With that said, it's by no means required, and a lot of books will teach you the LA that is required.*

As for Differential Equations, it's a very interesting and beautiful subject and to this day remains my favorite field in mathematics. In fact, if I wasn't addicted to having money, I would've done my PhD relating to Differential Equations. Anyway, at the introduction level, it isn't hard. You'll need some baisc LA, ie the concept of linear indepedence. Nevertheless, it isn't hard to learn it at the Boyce level, and that will get you started to solve a good number of equations you'll need to solve in Physics. So Analysis and Topology do not depend on Diffy Q. However, for more advance treatment of Diffy Q, you'll need Analysis and Topology (and this can be said about nearly every advacement treatment of any subject, including Probability!)

*Although if you do Spivak's manifolds, follow WannabeNewton's advice and know LA!

 

[subsonicman]

I guess you guys have convinced me to hold off on a comprehensive work on multivariable until after I've learned linear algebra. My plan now is to just get a general overview of the stuff I don't know in multivariable over winter break, using some easier book. While I love exploring math for its own sake, I don't want math to limit me in the physics classes I'm going to take.
I also recently found out my school has Apostol v.1 and v.2 as well as spivak calculus, so I will read those sporadically during next semester. Since I'm taking linear algebra next semester I can really delve into the multivariable books (be it Apostol, or the spivak manifolds book or something else) over the summer. Also I'll try to start Tenenbaum over winter break.

I'm sorry to ask so many questions, but you guys have been great so far. Do any of you know some relatively decent but easy calc/multivariable calc book to just fill in the gaps of my knowledge over the winter before I go more in depth over the summer?

 

[MarneMath]

http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/

If you want to just get an idea of what is covered in multivariable calculus and see how it may relate to single variable calculus, then this shall do well. The book focuses a lot of physical problems, which I think is good to be aware of. Sometimes, I think it's easy to get loss in the haze of mathematical rigor and not realize that some ideas can be extremely useful to real problems. Plus, this book has a solution manual provided, so you can check your progress as you go.

 

[Sankaku]

Yes, you want Linear Algebra before doing much in R^n. Look at the other threads here for recommendations.

However, don't let that stop you from having a go at it if you are enthusiastic (but don't start with Spivak's Manifold book). Two very good video courses (MIT and Berkeley) on MV Calculus:



http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/

 

[Cuauhtemoc]

I don't recommend Calculus on manifolds AT ALL for someone out of high school.
I think if you want to learn some rigorous calculus start with Apostol I, he also teaches you some linear algebra(that you will need for his second volume). Then Apostol II for multivariable calculus.
The good think about apostol is that you can get "all of calculus" by the same author, the same style...