What Are the Best Books to Deepen Your Understanding of Mathematics?

[Fragment]

To reply to Unknot's post, my aim is really to get as much as I can. Obviously I won't be reading 5 different calculus books in hope to get all I can get from calculus, so I'm really trying to find one or two books that can cover the necessary, and more.

Regards,

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[Werg22]

Also Apostol is the only one out of the three to cover DE's (in the first volume at least). In the 2nd edition, he also covers some linear algebra, but I think it's lousy, that's why I have the 1st.

 

[Fragment]

Did you find Apostol to be dull or never-ending? The way something is presented usually greatly helps out, and Apostol's book did not seem very "inviting".
I was also wondering, what about multivariate calculus? All three books don't seem to cover it really. Is Spivak's "Calculus on Manifolds" as good as his "Calculus"? (No, I'm not obsessed with Spivak, I just hear a lot of good about him)

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[Werg22]

I find Apostol's motivation of the integral the best from a pure mathematician point of view, and Courant's from an applied mathematician point of view. You shouldn't worry about multivariate calculus for now, and it's difficult to learn it properly without any linear algebra under your belt. I recommend either Hoffman or Friedberg, but maybe you should start with Lang (because the other two are more advanced and deal with abstract vector spaces and vector spaces in F^n where F is arbitrary, whereas Lang only does R^n). To learn multivariate calculus, there is no better learning source than Rudin's Principles of Mathematical Analysis.

 

[JG89]

Courant also covers Fourier series, which I don't think the other two do.

 

[Fragment]

Thanks a lot for the info guys. I had looked into Hoffman, seems very good, and has excellent ratings. I am convinced in Spivak, but I'm weighing out Apostol and Courant. Tell me, if you know, which one between Courant and Apostol provides the most practice/practice questions/topic assessment?

Many thanks,

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[qspeechc]

I read Courant/John and thought it was very good, very well motivated and explained. I do not think it is obsolete, people still read textbooks from 50 years ago, just think of Herstein. Also, Courant/John has nice applications, and remember, when you get to university, you will probably do a physics course too, where Courant and John will be handy.

I'm not sure if the differences in rigour between the books is as big a deal as people make it out to be. I mean, after all, you will study Rudin's PMA or some equivalent book later, right? And that's uber-rigorous.

Oh, as for the problems in Courant and John, there are plenty. None are plug-and-chug. Many challenge you to think harder. But, I have not read either Apostol or Spivak, so I can't compare.

 

[JG89]

I found Courant/John to be just as rigorous as Apostol. I can't compare it to Spivak though as I haven't read it.