# Calculus 2 book recommendation (exactly what I need inside)

## Main Question or Discussion Point

Next semester I will be taking Calculus 2.

I have used Courant and a little of Spivak for Calculus 1, and loved it! Extremely complete and challenging. Great books.

So, this is exactly what I will need to know:

Ordinary differential equations
Families of equations, initial value problems, separable equations, integrating factor, second order differential equation, boundary value problem, nonhomogeneous differential equation. Applications.
Real functions of several variables.
Graphs, isopleths, limits and continuity.
Partial derivatives.
Interpretation, higher order derivatives, tangent planes.
Differentials.
Linear approximation, increment theorem, chain rule, implicit differential (implicit function theorem).
Directional derivative and gradient vector, maximum and minimum values, Lagrange multipliers.

What would you guys recommend?

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Here are some nice books you should certainly check out:

Differential equations with applications - Simmons
Calculus on manifolds - Spivak
Analysis on manifolds - Munkres
Vector calculus, Linear algebra and differential forms - Hubbard

I would add to what micromass said:

Advanced Calculus of Several Variables - Edwards
Differential Equations and Their Applications - Braun

I also like ODE by Tenenbaum/Pollard, it's sometimes not very rigorous but contains all you need and is very clear!

Thank you for the help so far, micromass and alissca123

The only concern I have, is that those are many books. Do they all cover the topics I need? Can I simply choose one? I have little time and can't check them all out.

I heard the following in the past:
"If you know Calculus 1, you don't even need a book for Calculus 2"
and
"Spivak on manifold is much different than his Calculus 1 book, it is very dry and difficult to understand"

Do you guys have any opinion on this?

jbunniii
Homework Helper
Gold Member
Which Courant book did you use for Calculus 1? If it was "Introduction to Calculus and Analysis, Volume I", and you enjoy Courant's style, then take a look at volume II. I think it covers all of your listed topics.

Caution: Springer's edition of Volume II (the paperback, at least) is published in two separate books: part 1 and part 2.

Yes, I used Introduction to Calculus and Analysis, Volume I
But looking at Spivak it seemed also a good alternative.

Would you recommend Courant Volume 2 over the other books mentioned? If so, why?

jbunniii
Homework Helper
Gold Member
Would you recommend Courant Volume 2 over the other books mentioned? If so, why?
I would, for your purposes, because I believe it contains everything you need.

I don't think the Spivak book (assuming you mean Calculus on Manifolds) is an alternative, as it does not contain most of what you listed. In my opinion, it's also too concise, abstract, and unmotivated for a first exposure to the topics it does cover. It's not at all like his Calculus book in this respect. I don't see how a reader can have much appreciation for his treatment of Stokes' theorem on manifolds without first having a more concrete exposure.

I don't know the other books very well, although I have skimmed through the Munkres and the Hubbard books and they both seemed good. If I recall correctly, the Hubbard book in particular seems to be at a good level for someone who just learned calculus from the likes of Courant or Spivak. But I don't remember if it covers all or most of the topics you listed.

jbunniii
Homework Helper
Gold Member
So, it was basically rewritten with US students in mind.

However, I am neither European nor American, so for me it doesn't matter.

What version would you recommend? All I'm looking for is a DEEP understanding of the subject.

jbunniii
Homework Helper
Gold Member
I don't know the older version (Differential and Integral Calculus) at all. I ended up getting the newer version because I found an old hardback copy of volume 1 for \$20 in a used book store, and I liked it well enough to get the Springer paperback version of volume 2 (parts 1 and 2) to complete the set.

Thank you for the help so far, micromass and alissca123

The only concern I have, is that those are many books. Do they all cover the topics I need? Can I simply choose one? I have little time and can't check them all out.

I heard the following in the past:
"If you know Calculus 1, you don't even need a book for Calculus 2"
and
"Spivak on manifold is much different than his Calculus 1 book, it is very dry and difficult to understand"

Do you guys have any opinion on this?
All those books are great, so yes... you can simply choose one. But I think that you should skim through them and see which one fits your style.