# A book After Spivak: Calculus

1. Mar 8, 2013

### Astrum

I'm finishing up Spivak, after a break from "pure" math. I'm looking for a good follow up on multivariate analysis. I've heard that "Calculus on Manifolds" uses a lot of linear algebra (which I know very little of), so I'm on the look out for another suggestion.

I'm looking for a book on the analysis side, rather than computational, in the spirit of Spivak (which is rather challenging, yet rewarding!).

Suggestions?

2. Mar 8, 2013

### MarneMath

Courant volume II( Calculus) might be want you are looking for. I think you'll find a lot of similiarity between the way Spivak and Courant write and introduction material.

However, I do suggest you hold off on proof focused multivariate book, until you have a decent foot hold in linear algebra. I think, in many ways, the concepts form a more clear and concise picture when you can look at the subject from the lense of linear algebra.

3. Mar 8, 2013

### WannabeNewton

I doubt there exists a multivariate analysis book that doesn't heavily use LA. On the most basic level, arguably the most important theorem in multivariate analysis, the inverse function theorem, is itself one that relies on linear algebra.

4. Mar 8, 2013

### Astrum

Alright, so, I think I'll use Stewart for multivariate (computational).

Can you recommend a LA book? And after this, I can move on to a proof based text, correct?

By the way, I like the smell of Spivak's book.

Last edited: Mar 8, 2013
5. Mar 8, 2013

### Jorriss

I really like Axler, Linear Algebra Done Right. You can go to the math subtextbook forum to read reviews and opinions of it.

6. Mar 8, 2013

### micromass

Last edited by a moderator: May 6, 2017
7. Mar 8, 2013

### dustbin

x2. I think it will satisfy both your computational and theoretical needs. Beautiful book. Everything is well motivated and there are even many interesting applications. This book gets me incredibly excited about math every time I read it. If you include the appendix, you will be busy for a while.

Last edited by a moderator: May 6, 2017
8. Mar 8, 2013

### Astrum

I'll check this one out, thanks.
This is pretty expensive, is the 2nd edition really better than the first? I'd like to save some money, if possible.

9. Mar 8, 2013

### WannabeNewton

There is also the much more gentle but unfortunately very computational "Analysis on Manifolds" - Munkres that you can take a look at.

10. Mar 8, 2013

### Astrum

I'm really looking for a theoretical pure math approach to it. I've got a book (two in fact) for computational, which is good enough for physics, I suppose, I just happen to enjoy pure math.

I'm thinking of two different approaches. 1. Buy the all in one book, or 2. buy Linear Algebra Done Right, and I'll buy Spivak after.

Has anyone taken a look at Calculus on Manifolds from Spivak?

11. Mar 8, 2013

### WannabeNewton

Yes I am rather well acquainted with the book. The exercises are much better than those in Munkres but it isn't the best multivariate analysis book out there; it is rather unmotivated and is not as good as his amazing single variable calculus book. Still, it's better than Munkres in my opinion. There aren't many epsilon delta proofs in it which is a rather large disappointment since they are so fun, even in higher dimensions, but he has a lot of important elementary results regarding differential forms (and more importantly integration of forms which is so important in physics that you just HAVE to look at it from a pure math perspective). Micro's suggestion of Hubbard is probably the best at the level you are in considering Spivak makes use of a lot of linear algebra that he assumes the reader already knows.

12. Mar 8, 2013

### jasonRF

13. Mar 8, 2013