# What book should I get for multivariable calculus after Stewart?

• Calculus

## Main Question or Discussion Point

Hi.

I just finished the single variable part of Stewart's calculus book which helped me to master AP calculus. Now I am planning to move on to non-rigorous multivariable calculus. However, I have found reading his book a bit painful since the book mainly focuses on problem-solving techniques rather than the essence of calculus. For example, in the section on u substitution, instead of showing how it is related to the product rule of derivatives, he only teaches readers how to solve the problems so that they can pass their exams. This makes me want to switch to a new book for learning multivariable calculus.

I would like to be recommended a book written for building up intuition for multivariable calculus through a non-rigorous approach. I really like the way how Calculus with Analytic Geometry delivers the concepts to its readers; therefore it would be great if anyone can recommend me a book that resembles it. (You might be curious why I didn't get that book for my AP exam -- it is pricey.)

Thank you.

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## Answers and Replies

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mathwonk
Homework Helper
Have you gone to the library to browse books on the topic? Or is that not an option now? Lets see, have you tried Edwards and Penney? I myself only have rigorous books, and among those I like Courant (the least rigorous of my books), Wendell Fleming, and Lang.

The problem with several variable calculus is that it does not make sense conceptually unless you already know, or develop, some linear algebra. The point is that differential calculus is the technique of approximating non linear functions with linear functions, so you have to know about linear functions before you can begin. Notice they teach you about linear functions of one variable, namely y = mx+b, before you learn one variable calculus. But they do not always teach you linear functions of two or more variables before beginning several variable calculus, which is absurd. Then they are stuck trying to make do with one variable approximations to several variable functions, focusing on partial derivatives instead of the full derivative. This makes it virtually impossible to understand formulas like the several variable chain rule, much less the inverse and implicit function theorems. So if you want to understand several vbl calc, you really should study some elementary linear algebra first.

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TurboDiesel
Have you gone to the library to browse books on the topic? Or is that not an option now? Lets see, have you tried Edwards and Penney? I myself only have rigorous books, and among those I like Courant (the least rigorous of my books), Wendell Fleming, and Lang.
My city and its local university are at lockdown. I can't even find a place to get a haircut now.

I would love to read rigorous books on calculus once I enter the university. For now I need non-rigorous calculus to learn physics (E&M). Also, can I know what you think of Spivak's books?

Update: I searched the book you mentioned on amazon and I have to admit that the comments discourage me a little. Thank you anyway.

The problem with several variable calculus is that it does not make sense conceptually unless you already know, or develop, some linear algebra. Then point is that differential calculus is the technique of approximating non linear functions with linear functions, so you have to know about linear functions before you can begin. Notice they teach you about linear functions of one variable, namely y = mx+b, before you learn one variable calculus. But they do not always teach you linear functions of two or more variables before beginning several variable calculus, which is absurd. Then they are stuck trying to make do with one variable approximations to several variable functions, focusing on partial derivatives instead of the full derivative. This makes it virtually impossible to understand formulas like the several variable chain rule, much less the inverse and implicit function theorems. So if you want to understand several vbl calc, you really should study some elementary linear algebra first.
I learnt vector and a some 3D analytic geometry (dot/cross product, plane, line, normal, etc.) last year in my high school. It was very intuitive to me and I got nearly perfect on the exam. So I don't think it should be a problem.

I will watch video lectures and read a linear algebra textbook when learning it because someone told me I need to know some linear algebra concepts to understand Stroke's theorem.

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mathwonk
Homework Helper
which book discouraged you, and why, Courant? Edwards and Penney? Spivak's several variable calculus book (Calculus on Manifolds) is an excellent, highly rigorous, very brief (140 pages), very theoretical treatment of just the theoretical mainsprings of the subject, almost no applications at all.

Did you learn about linear maps in your linear algebra course? The topics you mention, (dot/cross product, plane, line, normal, etc.), are just the computational topics.

A very well regarded book on the subject of vector calculus including linear algebra is the classic by Williamson, Crowell, and Trotter, but I have not taught from it nor studied it myself.

Munkres also has a book, Analysis on Manifolds, that seems to be basically a rewrite of Spivak, hence presumably more readable. As far as price goes, I would not pay $50 for a paperback of these, and would go for a$5 used paperback of Williamson, Crowell, and Trotter.

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Leo Liu
bhobba
Mentor
The following is good if you want to understand rather than gloss over technical details:
http://matrixeditions.com/5thUnifiedApproach.html

It just assumes US Calculus BC or equivalent - but that is all. Beware though - it is rigorous - but done in a way that is not 'dry'. I know that you mentioned you would prefer something non-rigorous, but if you want something that goes beyond just teaching to pass exams some rigor can't really be escaped IMHO.

Another is the book by Boas - but it covers a lot more than just Multi-Variable Calculus - and is not cheap:
https://www.amazon.com.au/Mathematical-Methods-Physical-Sciences-Mary/dp/8126508108&tag=

I personally have both, and recommend both, but there is the cost factor.

Thanks
Bill

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rjurga and Leo Liu
Infrared
Gold Member
You might be interested in volume 2 of Apostol's calculus. It's more rigorous than a standard calculus book like Stewart, but not as theoretical as an analysis text. It reviews linear algebra and incorporates it as needed in the calculus portion of the text, as @mathwonk advised.

Disclaimed: I've never studied out of it, but I have a copy lying around, and it looks pretty good to me.

mathwonk
Homework Helper
I agree apostol is surely excellent, but I also have not studied it (I am however somewhat familiar with volume 1), and it is very expensive. I cannot consult it now as I gave my copy to the undergraduate math lounge library when I retired. Cost is the only reason I omitted mentioning it, as well perhaps as its high level of theory. But if you are considering Spivak, I second Infrared's suggestion of considering Apostol.

which book discouraged you, and why, Courant? Edwards and Penney?
Edwards and Penney. People on amazon think it's plain.
Did you learn about linear maps in your linear algebra course?
I think not. Well, it is not a linear algebra course -- it is a vector & differential calculus course.

Amazon has some sellers selling Apostol's books printed in India with affordable prices. But I think I am going to get a copy of Williamson's book for multivariable calc since I have to start with the first book of Courant or Apostol if I want to learn more rigorous calculus. Thanks.

vanhees71
Gold Member
2019 Award
My city and its local university are at lockdown. I can't even find a place to get a haircut now.

I would love to read rigorous books on calculus once I enter the university. For now I need non-rigorous calculus to learn physics (E&M). Also, can I know what you think of Spivak's books?
For that purpose, you can also get at least a start, reading the math parts of theoretical-physics textbooks. For vector calculus my favorites are

Sommerfeld, Lectures on Theortical Physics vol. 2 (hydrodynamics)
Abraham&Becker, Classical Theory of Electricity and Magnetism
Joos, Theoretical Physics

bhobba
mathwonk