# Need a proofy yet analytically inclined calculus book

## Main Question or Discussion Point

Need a "proofy" yet analytically inclined calculus book

I have James Stewart's Calculus book edition 4 that I use as a supplement, I really like it actually and I appreciate the analytic aspects of it. What I'm looking for is something that is both rigorous and analytic. And something at a very modest price also, preferably less than 30$. I hope I'm not asking for too much, thanks in advance! =D ## Answers and Replies Related Science and Math Textbooks News on Phys.org jgens Gold Member Exactly what do you mean by "analytic" here? For example: • Do you want a text emphasizing the computational aspects of calculus? • Do you want a text which addresses analytic functions? • Do you want an introduction to analysis type text? So before anyone can really give you good advice on text books, I think you will need to clarify your meaning here. As for the$30 price tag, you'll probably want to look at Dover books. This might help you narrow your search down a little bit.

Exactly what do you mean by "analytic" here? For example:
• Do you want a text emphasizing the computational aspects of calculus?
• Do you want a text which addresses analytic functions?
• Do you want an introduction to analysis type text?
So before anyone can really give you good advice on text books, I think you will need to clarify your meaning here.

As for the \$30 price tag, you'll probably want to look at Dover books. This might help you narrow your search down a little bit.
I'm sorry, I was very vague.

I meant calculus with analytic geometry.

I've looked at Dover, they have a terrible search engine and there are too many things to choose from so I'm feeling lost.

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jgens
Gold Member

I know Morris Kline wrote a calculus text which tries to emphasize some aspects of analytic geometry. He gives a lot of practical applications to things too, which may or may not be desirable. However, I don't remember his text being very rigorous. If I recall correctly, he proves most of the simple things (like rules for differentiation), but leaves most of the big theorems unproven. You can look at the book here: https://www.amazon.com/dp/0486404536/?tag=pfamazon01-20

If you're willing to spend a bit more on a text, Michael Spivak's Calculus is a very good exposition on the subject. The text is fairly rigorous and he makes a point of proving (almost) all of the results he introduces. He also has an appendix or two dedicated to some aspects of coordinate geometry, but it certainly isn't the emphasis of the text. You can looks at his book here: https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20

I know Morris Kline wrote a calculus text which tries to emphasize some aspects of analytic geometry. He gives a lot of practical applications to things too, which may or may not be desirable. However, I don't remember his text being very rigorous. If I recall correctly, he proves most of the simple things (like rules for differentiation), but leaves most of the big theorems unproven. You can look at the book here: https://www.amazon.com/dp/0486404536/?tag=pfamazon01-20

If you're willing to spend a bit more on a text, Michael Spivak's Calculus is a very good exposition on the subject. The text is fairly rigorous and he makes a point of proving (almost) all of the results he introduces. He also has an appendix or two dedicated to some aspects of coordinate geometry, but it certainly isn't the emphasis of the text. You can looks at his book here: https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20
Yes, I would much rather spend the money on Spivak; I suppose I need to save up some money. I'm just a little worried that I haven't been too involved in the art of proof. I realize that it requires a steep learning curve but I'm prepared to tackle the challenge.

It is too sad that they don't provide a preview for this book, so that I can get a feeling for what I'm getting.

I like Spivak but I think Apostal is better in terms of applications. However it has an "odd" order of presentation (integration before differentation).

jgens
Gold Member

I like Spivak but I think Apostal is better in terms of applications. However it has an "odd" order of presentation (integration before differentation).
Unfortunately, Apostal's text is much more expensive than Spivak's :(

It is too sad that they don't provide a preview for this book, so that I can get a feeling for what I'm getting.
You can always check on Google Books for a preview (3ed. - the other editions are very similar). I'd recommend you look beyond the first few pages as that is when it is much more rigorous. Try reading some pages further inside in the book (e.g. 250-252) if you want to get a feel for the style of the book. However, don't be suprised if you have no clue what he is talking about! You'll eventually know.

As for proofs, you could always try reading How to Prove It: A Structured Approach by Velleman. In my opinion, it was a great book. There's a preview feature on Amazon, but you can also try Google Books.

https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

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You can always check on Google Books for a preview (3ed. - the other editions are very similar). I'd recommend you look beyond the first few pages as that is when it is much more rigorous. Try reading some pages further inside in the book (e.g. 250-252) if you want to get a feel for the style of the book. However, don't be suprised if you have no clue what he is talking about! You'll eventually know.

As for proofs, you could always try reading How to Prove It: A Structured Approach by Velleman. In my opinion, it was a great book. There's a preview feature on Amazon, but you can also try Google Books.

https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20
Thank you! I like Spivak so far, and it starts out very simple yet rigorous. And that preview is very comprehensive compared to amazon's preview.

If you read Spivak, then I should warn you that its exercises are by far not easy. So don't feel discouraged if you find the exercises to be quite hard. It's perfectly normal.
I would certainly supplement Spivak with a less rigorous book with easier exercises. In calculus, you need to drill problems at times by making many of them. Spivak isn't really a good book for that.

I find that introduction to calculus and analysis by richard courant to be the best overall series, although spivak's calculus and calculus on manifolds texts will get the job done. Courant's is a more apostol like approach, but I didn't find it to be anywhere near as boring as apostol's