Good book on calculus?

[TubbaBlubba]

I've finished most of the calculus classes in high school, and they're... Lacking, to say the least. I'll probably study theoretical physics at the university, and I want to brush up and expand my calculus skills in advance.

I'm looking for a (or several, for that matter) book that treats what I guess is intermediate calculus, in particular differential algebra, various integration techniques, etc, preferably introducing multi-variable differentiation and the like. If it treats it in relation to physics or other natural sciences, all the better (I'm also studying Feynman's lectures on physics and some other college-level textbooks, and a better grip on calculus would be incredibly useful).

Suggestions for even further study is of course appreciated as well.

Thankful for any suggestions.

[W.D.Clinger]

I'm looking for a (or several, for that matter) book that treats what I guess is intermediate calculus, in particular differential algebra, various integration techniques, etc, preferably introducing multi-variable differentiation and the like. If it treats it in relation to physics or other natural sciences, all the better (I'm also studying Feynman's lectures on physics and some other college-level textbooks, and a better grip on calculus would be incredibly useful).


Michael Spivak, Calculus on Manifolds (http://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)), 1965.

[Ivor the Engineer]

For self study, Stroud's (http://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp/1403942463) books (http://www.amazon.co.uk/Advanced-Engineering-Mathematics-K-Stroud/dp/1403903123) are hard to beat.

[Vorticity]

These aren't really all specifically calculus-related, but just a few tips which I try give to every prospective physics major I meet:

1) Learn PDEs. A really good book is Partial Differential Equations for Scientists and Engineers (http://www.amazon.com/Differential-Equations-Scientists-Engineers-Mathematics/dp/048667620X). I love that book.

2) Learn probability theory. A good book is An Introduction to Mathematical Statistics and Its Applications (http://www.amazon.com/Introduction-Mathematical-Statistics-Its-Applications/dp/0321693949/ref=sr_1_1?ie=UTF8&s=books&qid=1291058574&sr=1-1).

3) Learn LaTeX (http://en.wikipedia.org/wiki/LaTeX).

Mastery of these three things will get you very, very far in physics.

[TubbaBlubba]

Michael Spivak, Calculus on Manifolds (http://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)), 1965.
Hmm, I think that wouldn't do me much good at the moment, a bit over my head. But thanks.
For self study, Stroud's (http://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp/1403942463) books (http://www.amazon.co.uk/Advanced-Engineering-Mathematics-K-Stroud/dp/1403903123) are hard to beat.
Oh, looks just like what I'm looking for. Great, thanks.
These aren't really all specifically calculus-related, but just a few tips which I try give to every prospective physics major I meet:

1) Learn PDEs. A really good book is Partial Differential Equations for Scientists and Engineers (http://www.amazon.com/Differential-Equations-Scientists-Engineers-Mathematics/dp/048667620X). I love that book.

2) Learn probability theory. A good book is An Introduction to Mathematical Statistics and Its Applications (http://www.amazon.com/Introduction-Mathematical-Statistics-Its-Applications/dp/0321693949/ref=sr_1_1?ie=UTF8&s=books&qid=1291058574&sr=1-1).

3) Learn LaTeX (http://en.wikipedia.org/wiki/LaTeX).

Mastery of these three things will get you very, very far in physics.

Thanks, I'll look at those two books, and I know a bit of LaTeX from use at fora, really useful.

[TubbaBlubba]

I ordered the books Ivor linked to, Partial Differential Equations..., as well as Introduction to Mathematical statistics... though I picked the fourth edition since it didn't have a waiting time of 2 months and was half the price.


I then call them "Yule gifts" and make my parents pay for them.

[Drachasor]

Spivak is good, but I recommend this book:

http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896

[Jorghnassen]

All calculus books are mediocre. Even Stewart.

[TubbaBlubba]

Spivak is good, but I recommend this book:

http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896

Jesus, that looks like the lovechild of a bible and a brick. I think I'll pass.

[Drachasor]

Jesus, that looks like the lovechild of a bible and a brick. I think I'll pass.

It is the best introductory Calculus book I've ever seen. If you want a rigorous understanding of Calculus, that's the way to go.

That said, a textbook isn't great without being able to practice. You should get a book of problems and solutions in basic calculus. You could also buy a solutions manual for regular text, but that's expensive.

Edit: FYI, you are literally judging a book by its cover there.

[TubbaBlubba]

It is the best introductory Calculus book I've ever seen. If you want a rigorous understanding of Calculus, that's the way to go.

That said, a textbook isn't great without being able to practice. You should get a book of problems and solutions in basic calculus. You could also buy a solutions manual for regular text, but that's expensive.

Edit: FYI, you are literally judging a book by its cover there.

I'll consider it later. (also, ordering from the US is expensive). I've got some material to work with at the moment anyway. I'm still open to suggestions, beyond the ones I already bought, by the way.

[TubbaBlubba]

Aight, got my Partial Differential Equations and [Advanced +] Engineering Mathematics, still waiting for Statistics.

But damn, these [EM/AEM] are BOOKS, about 1200 pages each.

[Furcifer]

All calculus books are mediocre. Even Stewart.

I've had about 3 different editions of Stewart, they seem to be the standard around here. I agree, they all see to be mediocre, but I find these tend to have more examples. They can get worse, but they don't seem to get much better.

[sol invictus]

Aight, got my Partial Differential Equations and [Advanced +] Engineering Mathematics, still waiting for Statistics.

But damn, these [EM/AEM] are BOOKS, about 1200 pages each.

There's no good reason for an introductory calculus book to be so long. For differential equations there's some justification for it, because there is no one method for solving them (which means there are many techniques, each of which works only in a few special cases, plus lots of tricks for solving them numerically).

The most important thing is to understand intuitively what you are doing at each step. Many calculus books I've seen focus on techniques of integration, and have you perform hundreds of integrals by various tricks. In my opinion this is far from the optimal way to learn the subject - it's roughly equivalent to learning how to perform square roots by hand, and teaches you about as much (which isn't nothing, but isn't much either). Any integral you can do by hand you can look up or have a computer calculate for you.

Instead, if there are books that take a more formal approach (construct the real numbers from the field axioms, define functions carefully, etc.) I'd focus on those. Unfortunately I can't recommend one, because I never used a book.

For theoretical physics you could also benefit from studying more fun types of math - linear algebra to start, then group theory and group representations, topology, and differential and algebraic geometry.

[TubbaBlubba]

There's no good reason for an introductory calculus book to be so long. For differential equations there's some justification for it, because there is no one method for solving them (which means there are many techniques, each of which works only in a few special cases, plus lots of tricks for solving them numerically).

The most important thing is to understand intuitively what you are doing at each step. Many calculus books I've seen focus on techniques of integration, and have you perform hundreds of integrals by various tricks. In my opinion this is far from the optimal way to learn the subject - it's roughly equivalent to learning how to perform square roots by hand, and teaches you about as much (which isn't nothing, but isn't much either). Any integral you can do by hand you can look up or have a computer calculate for you.

Instead, if there are books that take a more formal approach (construct the real numbers from the field axioms, define functions carefully, etc.) I'd focus on those. Unfortunately I can't recommend one, because I never used a book.

For theoretical physics you could also benefit from studying more fun types of math - linear algebra to start, then group theory and group representations, topology, and differential and algebraic geometry.

The books are much, much more comprehensive than that, they represent the typical first and second years of math studies one who studies science or engineering would attend to, I think.

[Vorticity]

Aight, got my Partial Differential Equations and [Advanced +] Engineering Mathematics, still waiting for Statistics.

But damn, these [EM/AEM] are BOOKS, about 1200 pages each.

Well, the PDE book I recommended above isn't so brick-like. It's around 400 pages.

Of course, you'll want to make sure you're good on calculus before moving on to PDEs...

[TubbaBlubba]

Well, the PDE book I recommended above isn't so brick-like. It's around 400 pages.

Of course, you'll want to make sure you're good on calculus before moving on to PDEs...

Yeah, I got that too, I think it'll end up on the shelf for a while.

[Ivor the Engineer]

Free mathematical software TubbaBlubba and others may be interested in:

Euler Math Toolbox (http://eumat.sourceforge.net/download.html) is an excellent little program with a Matlab-esque frontend and an interface to the symbolic mathematical package Maxima (http://maxima.sourceforge.net/).

For numerical computing and simulation, Scilab/SciCos (http://www.scilab.org/) are very similar to Matlab/Simulink.

While both the above programs can handle statistics, R (http://www.r-project.org/) has better support for performing statistical tests on tables of data.

[TubbaBlubba]

I've been playing with Euler Toolbox a bit before, actually, after looking over the software in the Linux Package Handler. Very nice program. Thanks for the links.

[Chris Haynes]

The books are much, much more comprehensive than that, they represent the typical first and second years of math studies one who studies science or engineering would attend to, I think.

For some more general historical and recreational treatment of math, try the books by Paul Nahin (http://www.amazon.com/Paul-J.-Nahin/e/B001HCS1XI/ref=ntt_athr_dp_pel_pop_1). They do stretch from algebra (http://www.amazon.com/Imaginary-Tale-Princeton-Library-Science/dp/0691146004/), through calculus (http://www.amazon.com/When-Least-Best-Mathematicians-Discovered/dp/0691130523/) (which my avatar is reading!) and onward (http://www.amazon.com/Dr-Eulers-Fabulous-Formula-Mathematical/dp/0691118221/). He is a retired professor of electrical engineering who has decided to write about math. He claims he has readers who always have a piece of paper and pencil nearby.

[TubbaBlubba]

For some more general historical and recreational treatment of math, try the books by Paul Nahin (http://www.amazon.com/Paul-J.-Nahin/e/B001HCS1XI/ref=ntt_athr_dp_pel_pop_1). They do stretch from algebra (http://www.amazon.com/Imaginary-Tale-Princeton-Library-Science/dp/0691146004/), through calculus (http://www.amazon.com/When-Least-Best-Mathematicians-Discovered/dp/0691130523/) (which my avatar is reading!) and onward (http://www.amazon.com/Dr-Eulers-Fabulous-Formula-Mathematical/dp/0691118221/). He is a retired professor of electrical engineering who has decided to write about math. He claims he has readers who always have a piece of paper and pencil nearby.

Oh, looks neat. I'll look into it.