0

Okay, this should be a quick and easy question for those of you who've studied calculus. I have a list of books that I want to order by topic, the books are as follows:

  1. Michael Spivak - Calculus
  2. Michael Spivak -
  3. Tom Apostol - Calculus, Vol. 1
  4. Tom Apostol - Calculus, Vol. 2
  5. Richard Courant Differential and Integral Calculus, Vol. 1
  6. Richard Courant Differential and Integral Calculus, Vol. 2
  7. Morris Tenenbaum - Ordinary Differential Equations
  8. Tom Apostol - Mathematical Analysis
  9. James R. Munkres - Analysis On Manifolds
  10. Michael Spivak - Calculus On Manifolds
  11. Morris Tenenbaum - Ordinary Differential Equations
  12. Richard Haberman - Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems
  13. Robert Strichartz - Guide to Distribution Theory and Fourier
  14. Loomis, Sternberg - Advanced Calculus
  15. G. H. Hardy - A Course of Pure Mathematics

No I'm not planning on studying all these books, however I do feel it would be nice to have a feel of what's where, and if I do get stuck on one book or the other it would be perhaps be nice to dip into another and see if the exposition is better or see how the expositions differ. At the moment I'm trying to build a mental roadmap of this topic in my head (It's what works for me).

From my very limited knowledge I believe that calculus is ordered and roughly broken up as follows:

  1. Calculus of one-variable
  2. Calculus of several-variables
  3. Ordinary and Partial Differential Equations
  4. Introductory Analysis? Rigorous Calculus?
  5. Analysis (However I don't believe very many of the above books are real analysis books)

If this list of topics could be better please amend it.


So what I need help with is classifying these books using those topics and these question:

  1. Is calculus of several variables the same topic as multivariable calculus and is that the same as vector calculus?
  2. Is the topic of applied differential equations a sub-topic of Ordinary Differential equations?
  3. Where do partial differential equations fit within this all? Is it it's own topic?
  4. What is advanced calculus? Is it introductory analysis or what?
  5. Are there any other canonical books you feel should be added to this list?
  6. What should I study after Spivak's Calculus?
seeker
  • 6,358
  • 10
  • 68
  • 101
  • 1
    Not a direct answer, but a helpful resource, is the taxonomy given here: http://www.math.niu.edu/~rusin/known-math/index/index.html – HTFB Dec 22 '14 at 15:42
  • @Bye_World Calculus on manifolds is on the list. Also when would you it is best to learn Linear Algebra? After spivak's calculus? – seeker Dec 22 '14 at 15:48
  • 2
    Learn linear algebra either concurrently with calculus or immediately afterward. Don't start on multivariable or analysis without having had linear algebra. –  Dec 22 '14 at 15:53
  • @seeker If you'd like an idea of the order that these topics should come in, you could take a look at the (incomplete) answer I gave to [this question](http://meta.math.stackexchange.com/questions/18946/learning-roadmap-request-compiling-a-mathematics-stack-exchange-undergraduate). If you like it you might want to take a picture of it, though, because that question will likely be closed soon. –  Dec 31 '14 at 00:08
  • @Bye_World thank you! That's really useful! – seeker Dec 31 '14 at 00:19
  • 1
    I don't think Spivak's *Calculus on Manifolds* is good to learn from. It's good if you want a quick exposition of material you already mostly know. What you study after Spivak's book depends partly on whether you have a particular field of application in mind. If it's only math, then have a look at the bibliography at the end of Spivak's book after you're done reading it. If you don't know what you want to read next, then the default choice would be algebra. For people interested in pure math, it's best to combine linear algebra and abstract algebra in the same exposition. Artin's book... – user204305 Jan 09 '15 at 06:13
  • 1
    ... *Algebra* is great, in my opinion. In fact, if you read the first four chapters of Spivak's book (3rd edition) and find that it's not excessively difficult for you, then it ought to be possible to start Artin's book then, if you're interested. *Clarification*: Above I meant after Spivak's *Calculus*. – user204305 Jan 09 '15 at 06:18
  • @user204305 I've read Artin's book is extremely rigorous and wouldn't be such a great first exposition to the subject, do you have any other algebra books to recommend? When is one ready for Artin's Algebra, what are the prerequisites? – seeker Jan 09 '15 at 09:58
  • In terms of factual knowledge, there are no specific prerequisites for Artin's book, other than knowing what the real numbers are. It helps, but isn't strictly necessary, to have seen vectors. It does require some "mathematical maturity", but as I said, if you don't have any trouble with the beginning of Spivak, you should be fine with Artin. If Spivak's book is too difficult, then read Apostol's *Calculus* instead. Then re-evaluate after most of the first volume what you want to do next, perhaps a linear algebra book, or even just get your linear algebra from Apostol. – user204305 Jan 09 '15 at 14:27
  • A correction. For Artin's book, you need to know about complex numbers and induction. He writes that you should know calculus too, but this would only be in isolated spots. – user204305 Jan 09 '15 at 14:45
  • Where do you draw the line between 'Introductory Analysis' and 'Analysis'? – John Gowers Apr 28 '15 at 14:54

1 Answers1

3

I haven't studied all of these, so I'll make this a CW so others can edit in the rest (if they feel like it).

Calculus (just a good calculus reference book, because you don't necessarily need rigor when you're first starting out):

  • Calculus by Ron Larson & Bruce Edwards

"Rigorous" Single-Variable Calculus (AKA calculus with some analysis):

  • Calculus by Michael Spivak
  • Calculus, Vol. I by Tom Apostol (note that Tom's books also cover linear algebra)
  • Differential and Integral Calculus, Vol. I by Richard Courant

Ordinary Differential Equations:

  • Ordinary Differential Equations by Morris Tenenbaum
  • Ordinary Differential Equations by Vladimir I. Arnold

Partial Differential Equations:

  • Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems by Richard Haberman

Real Analysis:

  • Understanding Analysis by Stephen Abbott
  • Mathematical Analysis by Tom Apostol
  • Principles of Mathematical Analysis by Walter Rudin
  • A Course of Pure Mathematics by G.H. Hardy

Multivariable Calculus (just a good first multivariable calculus book, because you don't necessarily need manifold theory when you're first time learning multivariable):

  • Vector Calculus by Jerrold Marsden & Anthony Tromba

"Rigorous" Multivariable Calculus (AKA multivariable with some analysis and other stuff):

  • Calculus, Vol. II by Tom Apostol
  • Differential and Integral Calculus, Vol. II by Richard Courant
  • Advanced Calculus of Several Variables by C.H. Edwards, Jr.

"Rigorous" Multivariable Calculus with some Manifold Theory (Intro to Differential Geometry):

  • Calculus on Manifolds by Michael Spivak
  • Advanced Calculus: A Differential Forms Approach by Harold Edwards

Analysis on Manifolds:

  • Analysis on Manifolds by James R. Munkres

Some weird combination of Calculus, Real Analysis, Manifolds, Linear Algebra, and Classical Mechanics:

  • Advanced Calculus by Shlomo Sternberg & Lynn Loomis

NOTE: Just because some books are listed in the same category above does not mean that they are at the same level or cover exactly the same topics. Some of the books above are very different from their neighbors. If you need help choosing a textbook for self-study, I'd recommend asking your professors -- they will have a better idea of what exactly you already know and what exactly you'll need to know.

NOTE 2: Neither the categories nor the books within the categories in the above are ordered in terms of difficulty.


Answers to your questions:

Is calculus of several variables the same topic as multivariable calculus and is that the same as vector calculus?

They all mean the same thing, though not every book on this topic will be at the same level.

Is the topic of applied differential equations a sub-topic of Ordinary Differential equations?

There are two subfields of differential equations: ordinary differential equations (ODEs) and partial differential equations (PDEs). Most texts on differential equations will be highly applied because that's the origin of most of the interesting problems of the subject.

Where do partial differential equations fit within this all? Is it it's own topic?

It is a separate topic from ODEs. ODEs are about solving differential equations for functions of one variable, while PDEs solves for functions of several variables.

What is advanced calculus? Is it introductory analysis or what?

A book called "Advanced Calculus" could have several meanings. Often it is a blend of multivariable calculus and analysis, where analysis is basically just "rigorous" calculus with a little bit of the theory of metric spaces.

Are there any other canonical books you feel should be added to this list?

I've added a couple, but this is really too many topics for anyone to make a comprehensive list.

What should I study after Spivak's Calculus?

If you haven't taken linear algebra, yet, that should be your next topic. If you have, then multivariable calculus (possibly Spivak's Calculus on Manifolds), Real analysis, or ODEs could come next. Or, if you aren't set on calculus/ analysis, you could go on to Lie theory (a la Stillwell's Naive Lie Theory), abstract algebra, probability theory, geometry, Clifford algebra, Combinatorics/ Graph Theory, or Elementary Number Theory. You have a lot of choices once you've gotten the basics (high school math, calculus, and linear algebra) out of the way.

  • Hostetter is now out of the trio on Larson and Edward Calculus? – dustin Dec 24 '14 at 18:50
  • He is in my copy. –  Dec 24 '14 at 18:52
  • He is in my copy too but the new edition doesn't list him. I only looked since you only listed Larson and Edwards. – dustin Dec 24 '14 at 18:53
  • 1
    Sorry, I meant that he is *NOT LISTED* in my copy. I have the $9$th edition here. –  Dec 24 '14 at 18:54
  • @Bye_World another question sorry, what is calculus of variation? And where does it fit? – seeker Dec 31 '14 at 00:20
  • 1
    @seeker Calculus of variations is the study of optimization of functionals. If you're studying physics, you'd be interested to know that it is the basis of analytic mechanics. If not, I'd say don't worry about it until/ unless you need it. –  Dec 31 '14 at 00:22
  • I think of vector calculus as relating specifically to grad, div and curl. – user204305 Jan 09 '15 at 06:08
  • 1
    @user204305 The [Wikipedia article](http://en.wikipedia.org/wiki/Vector_calculus) would agree would you, but it does mention that `The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus`. So if you'd like to edit the post, you can. I specifically made this a community wiki. –  Jan 10 '15 at 00:12
  • @Bye_World how would you order topics such as: linear algebra; vectors and matrices; groups, field rings; abstract algebra ? – seeker Jan 11 '15 at 10:36
  • 1
    @seeker Vectors and matrices are subtopics of linear algebra and groups, fields, and rings are subtopics of abstract algebra. I'd learn linear algebra first, but if you wanted you could just learn the basics of vectors and matrices first and then choose an abstract algebra book that had a pretty good section on linear algebra. –  Jan 11 '15 at 16:31
  • @Bye_World thanks again! – seeker Jan 11 '15 at 17:42
  • @Bye_World I just wanted to drop in to thank you again, your help has been very valuable to me, thank you! – seeker Apr 28 '15 at 14:45