Self teaching Multivariable calculus

[p53ud0 dr34m5]

hello, i was wanting to teach myself multivariable calculus. i am currently in calculus BC AP. that class isn't challenging enough for me. do any of you out there know any good sites or good books for multivariable calculus. any help would be greatly apprecaited.

thanks in advance.

 

[philosophking]

Your math teacher would probably have some good resources for you. Since your AP's in May, I suggest you concentrate on BC until you take the test (just humor me, hehe), and then afterwards ask your math teacher if he or she has any books that she'd like to lend you.

 

[Crosson]

Try Stewarts "multivariable calculus", it is good for teaching yourself. Also, consider exploring differential equations.

 

[p53ud0 dr34m5]

i was thinking about purchasing that book. i saw it on eBay. heh...
as for concentrating on BC...i knew it before i entered the class, because i taught myself calculus during my 9th grade summer... and my school is gay so they wouldn't let me in bc in 10th grade. but thanks for the suggestions guys. any more? id appreciate all suggestions.

 

[tornpie]

If you are just studying for the AP test, any old calc text will probably do.

If you want to learn math the right way, consider getting Spivak's Calculus. Then you should be ready for Rudin's Principles of Mathematical Analysis and Spivak's Calculus on Manifolds. Also consider looking into linear algebra. I recommend Strang's text and the video lectures available online at

http://ocw.mit.edu/OcwWeb/Mathematics/18-06Linear-AlgebraFall2002/VideoLectures/index.htm

After this, you can go on to Hoffman/Kunze's Linear Algebra or Sheldon Axler's Linear Algebra Done Right. The second one is much more pedagogical and clean. The first one is the traditional Linear Algebra text and contains a bit more material.

Anyhow, don't let your school determine (and in the process ruin) you math education. Use a combination of those books and places like this.

 

[p53ud0 dr34m5]

well, i already know calculus. i don't have to study for the AP exam. my teacher said i should get a 5 on it...heh, she maybe a little TOO confident. i am really looking for material on multivariable calculus. well, i would really like anything that is above calculus BC. that should broaden the topic a tad.

 

[tornpie]

It depends on what you want to use your math for eventually. If you want to be a mathematician (or semi-competent physicist) start looking into rigorous calculus books like Rudin (or Bartle and Sherbert for a less steep learning curve). Otherwise, any cheap text will probably do.

 

[fourier jr]

maybe start on some analysis & proving things. check out rudin's principles of mathematical analysis or pfaffenberger/johnsonbaugh's foundations of mathematical analysis

 

[mathwonk]

stewart is a mass market book for average students, but probably bettter written than many. spivak's "calculus on manifolds" is outstanding but assumes linear algebra first.

the best classical book, and one i would recommend to you is courant's calculus volume 2. another outstanding book is apostol's volume 2. the problem is, even if you "know" calculus, where did you learn it? you could have difficulty with a high level calculus book on several variables if you learned out of a cookbook for one variable, like stewart.

that's why you are being recommended to get spivak's one variable calculus book, or courant's or apostol's volume 1, and relearn that material right.

you sound like someone for whom a good book would be appropriate. unfortunately the whole AP program is a disservice to students like you, who then skip getting a decent course on calculus n college from someone who actually understands the material much better than most high school teachers.

some of them are gay though. was that a misprint, or are you a homophobe?

 

[CrankFan]

spivak's "calculus on manifolds" is outstanding but assumes linear algebra first.

Spivak also have a book called Calculus. I wonder how this compares with some of the other books mentioned.

 

[mathwonk]

I mentioned it above in this sentence:

" that's why you are being recommended to get spivak's (one variable) calculus book, or courant's or apostol's volume 1, and relearn that material right."

 

[courtrigrad]

mathwonk I currently have been working from Courant's book because I am one of those people who hate "memorize this rule and solve." Do you recommend reading the book like a story book sequentially, or skipping around? Should I do every single problem including the Appendix problems and Miscellaneous Problems? How did you fare with your classes with this book?
Thanks

 

[mathwonk]

i have myself never been able to read any book all the way through, except maybe a riveting spy book, like one i read once about the second world war.

in courant i have found certain sections that i can read through and get some little nugget of wonderful stuff out.

like the appendix in volume one where he explains about the concept of the point of accumulation. i still remember that after 45 years. and the section in the beginning where he shows the relation between decimals and points on the line.

and a wonderful section, maybe at the end of the second volume, or first volume where he shows how path integration of 1/z gives an explanation of the multivaluedness of the logarithm, according to which route around the origin the path takes.

or the little section at the beginning of the second volume where he explains about the formula for volume of a parallellepiped, using determinants.

or the section in vol2 where he explains clearly the meaning of a derivative of several variables.

just take what you want. do as many problems as you can.

enjoy it. if you set yourself the task of reading it all in order, or not allowing yourself to enjoy the end before the beginning, you run the real risk of never going anywhere with it and giving up completely.

of course there are exceptional people like some of my friends who just plowed right through a big book, and did all the execises. and some of them are famous now, some not.

i have also beneifted from time to time by finally reading something basic that i should, but trying to read everything cannot be allowed to become an excuse for not reading anything.

this book may have been too hard for most of my classes. i did not get back my evaluations for last semester yet, when we used it.

i know that. but i would rather blow some people away than cheat the really ambitious people. also even those who are blown away now, stand to learn the stuff in future, more so than if they never saw it done well at all.

some students erroneously think a class is about getting a good grade now, and i think it is about exposing them to powerful ideas that will grab them and never let go until they bear fruit.

when i took calc, i was blown away, or thought i was. then i compared notes at christmas with a friend who went to a well known engineering school. he did not know half of what i had learned. i went back encouraged to learn more, realizing they had higher standards where i was going.

 

[CrankFan]

I mentioned it above in this sentence:

" that's why you are being recommended to get spivak's (one variable) calculus book, or courant's or apostol's volume 1, and relearn that material right."

:uhh:

So you did, so you did... I have an excuse but I'll spare you

 

[courtrigrad]

thanks a lot. Frankly if I did every single problem in the book, it would take years for me to finish. What I am doing is skipping to other sections, really understanding them, and doing a couple of problems. Is this discontinous, or would you say that this is allowable? Because Schwarz inequality and all of the stuff in the beginning are not important for me right now. I first want to glean information about the important topics, morever enjoy it. Do you think this is a reasonable way to study Courant?

Thanks a lot

 

[mathwonk]

sure read whatever you want. i always skipped that horrible section on cauchy schwartz myself.

if you want to learn calculus though eventually you need to read about continuity, pages 46-56, and integration 76-86, and differentiation, and their connection, on up through page 120.

as for cauchy shwartz, it follows from the law of cosines, which itself follows from the pythagorean theorem, that if v and w are vectors then v.w = |v| |w| cos(t), where t is the angle between them, and v.w is their dot product. hence since |cos(t)| is never greater than 1, we get |v.w| <= |v| |w|. that's the c-s inequality.

apparently we are talking about vol 1 here.

 

[courtrigrad]

yes. I plan to finish vol 1 within a week and go to vol 2. I am reading about the important topics.

Thanks a lot

 

[devious_]

mathwonk, how does, in your opinion, Courant compare to Apostol?

 

[mathwonk]

well i like em both very well. apostol as i recall is more precise, having been written 30 years later, in the era of careful mathematical writing, the 1960's, whereas courant was a pioneering text from the 30's and is by a guy who is also a writer on mathematical physics. so it has more applications. but i think apostol has some of those too. apostol is more meticulous and dry, courant has more charm.

so courant is not by modern standards as "rigorous" as apostol, but the essentials are there.

apostol is incredibly scholarly and careful, and clear too. both of them start with integration instead of differentiation, the correct historical ordering.


gosh I cannot pick one over the other, but I'm leaning towards courant.

to be honest though, as a young student, courant was hard for me to get far in.

i did not realize i should read it in palatable pieces, and got discouraged by not making it through the dense parts.

apostol can do that to you too.

as a teacher though, i was amazed at how carefully apostol covered all the bases.

i myself have seldom learned anything straight out of one book, at one session. everything has to go around and around for me, and settle out, in ways that are partly psychological.

so I use more than one book.

young people really like spivak though. he works hard to make it appeal to the very bright, but also naive young student.

i actually learned much of calculus for the first time out of spivak, while grading the course as a grad student.

then years later as a teacher i realized that the same stuff was in courant, and that spivak had just cleaned it up, and repackaged it.

then more years later, i read apostol and was again impressed that i was still learning a lot i still did not know.

you cannot go wrong with any of those books. but don't feel left out if they do not suit at first encounter.

shoot, my best math buddy at harvard, taking the elite honors calc class said his favorite book was silvanus p thompson's calculus made easy, and i like it too.

it took me years to realize that the down home funky stuff thompson says is actually right though, because he didn't prove anything. i like proofs.

stewart can be useful too, and thomas, or edwards and penney. i sniff at some of these as cookbopoks, but I am willing to learn anywhere i can.

i get a little out of each source, whatever that source does well.

there is no oine source in general that does everything best.


(I might make a one or two exceptions in my own very specialized field of research, as there are a couple of experts who have written some great texts on things they excel everyone at.)

but the perfect calculus book does not exist.

I like Spivak, because it spoke clearly to me, and allowed me to make the amterial my own, so that I no longer need to refer to the book. When someone gets beyond a book, and begins to pooh the very book he used to value, that isa big compliment to that book. i.e. the book gave him all he needed and allowed to amke it his own, so that he no longer needs the book. then the book falls away and becomes superfluous. that is a good book.

spivak did that for me.

but i got some ideas from courant and some from apostol.

So it is one thing to say how scholarly a book appears to an old person, and another to say which one a young person can best learn from. that is a personal matter. One of my friends however who is a professor of topology learned very well from apostol as a student at MIT. Spivak was used at Chicago recently, and Courant was used in the 60's at Harvard.

I myself own them all, as well as Joseph Kitchen's fine book, and also G.H Hardy's classic, Pure Mathematics, and also Jean Dieudonne's Foundations of Modern Analysis, and also Lynn Loomis' Advanced Calculus, and also Serge Lang's Analysis I and II. I especially like Lang's books. They are so clear, and to the point.

I also own Calculus Made Easy, Stewart, Edwards Penney (6 different editions), Lipman Bers book, Thomas (several editions), and am recently trying to incorporate the ideas of lebesgue integration into my grasp of calculus, from books like Rudin's Real and Complex Analysis, and Riesz Nagy's Functional Analysis.

 

[devious_]

Thanks for the insightful response. I really appreciate it.

I'm currently using Apostol and I'm find the first few chapters a bit dry. I've already studied the topics of the first 10 chapters in school, but I intend to go through them quickly to learn the basics in a more rigorous manner before proceeding to the chapters I know very little about. Hopefully it won't be too difficult.

I'm finding the proofs of seemingly 'trivial' things tedious and hard -- like, for example, proving that the ordinate set of a nonnegative bounded function is measurable and its area equals its integral. I realize that this is an important result, but I'm really not enjoy proving this like this analytically, if you know what I mean.

I'll just trudge along in the meanwhile and wait till I graduate and go to college, where hopefully I'll have access to several books and get a wider point of view. :)

 

[mathwonk]

huh? what is the "ordinate set"? is that the region under the graph?

what is his definition of measurable? I didn't know there was measure theory in there.

I don't know that stuff very well, but maybe if you tell me the definitions I can make it seem easier for you.

But of course it should be true that the integral of a positive function equals the area under the graph.

so a result like that just says you have the right definition of integral, or the right definition of area, one or the other, or at least that they are compatible.

you see my attitude now is that I do not care what they say in detail so much, I care about what point they are trying to make, and how well they make it.

When I started learning math I though what they said was gospel and i should memorize it. Now I realize they are trying to address a problem, and I want to understand what the problem is, what their approach to it is, and think about whether their approach is reasonable, and if not, come up with a better one myself.

Then when I do so, I almost always discover it has already been done by someone, but then I have a way to distinguish between books that are thoughtful and books that merely copy what they read in some other book, without thinking logically about what they are doing.

are you using apostol's "mathematical analysis"? Thats the advanced calculus book. he also has a beginning calculus book, in 2 volumes.

 

[mathwonk]

for example what is the integral of a non negative bounded function?

well the lower integral is just the "least upper bound" (smallest number not smaller than) the area of all families of rectangles that lie under the graph.

now what is the measure of that set? well certainly the measure of a family of rectangles is its area, and measure should go up for bigger sets, so definmitely the measure of the region under the graph should be at least as large as the area of all the rectangles under there, i.e. at least as large as the integral.


Now the only way the measure could be larger than the integral, is if its definition allows some more refined approximation of that region, i.e. by sets that fit better than rectangles.

but now we look at the upper integral, which is the greatest lower bound of all areas of rectangles that lie above the region. And by definition of integral, if it exists, then that number agrees with the lower integral. Also it seems the same argument I gave before shows that the measure of those upper rectangles agrees with their area, so the measure should also be no larger than the upper integral.

so the measure should be caught between the same two numbers that define the integral since they are equal, all the numbers are equal.

is this too tedious?

you see all you need for this proof is the monotonicity of integrals and also of measure, and the fact they agree on rectangles. that's all.

the proof of these facts involves the details of the definitions, which you notice i did not even give here for measure. does this help?

 

[mathwonk]

oh by the way, I have noticed the word "quickly" popping up in several posts by peopel saying how they were going to read apostol or courant. I may be going out on a dnagerous limb here but I must say, that is ridiculous.

No one reads Courant or Apostol "quickly", say in a few weeks, and learns anything well. It is worth hours and sometimes days spent on a single page with real ideas in it.

The best teacher I ever had (and one of the smartest researchers and people generally) said: "if you do not write at least 5 pages per page of mathematics you read, then you have not learned anything". He said the only person he ever met who did not have to do this was Paul Cohen, a Fields medalist.

So to read apostol that way you would have to write at least a couple thousand pages of scratch work.

Are you doing that?

 

[mewmew]

Thank you!

Thank you! I am reading up on Courant after trying Spivak and I must say that the easiest things are usually the hardest to prove, or atleast the hardest for me to understand. When you mentioned it taking days to understand a page it made me feel a lot less dumb because my non math friends always make fun of me when I look at one page for a day or two and constantly just scrible thoughts and ideas about it down. These books take some heavy mental power(well, atleast from me they do!)

 

[courtrigrad]

yes i meant i was going to spend a week on one topic. plus i havea whiteboard so i write down my ideas there

thanks

 

[devious_]

I have volume 1. In the first integral calculus chapter he defines a measurable set as a set to which an area can be assigned, and an ordinate set is basically the area under the curve.

I understand the concept fine, it's just that I'm not really comfortable proving things like this. I guess it's because I'm new to proving almost everything. I just need some time getting used to it.

Oh, and when I said quickly I meant a month or so. I already know almost everything in the first 10 chapters, so I'm basically reviewing and learning a couple of new ideas here and there, and when I do learn something new, like a proof, I try to reproduce it after I finish reading a page, that's the only way I can ensure it stays in my memory. So I do intend to use a lot of scratch paper, especially when I finish the first 10 chapters.

 

[mathwonk]

what is his way of assigning area? you said he was proving that the area under the graph was the integral. to porve that you need seaparate definitions of area and integral.

or did he just give axioms for area as I did above?

 

[devious_]

Yeah, he gave 6 axioms.

 

[JonF]

I learned all of my calculus from Stewart’s book. I thought it did a pretty good job.

 

[mathwonk]

I just looked at the proof that the integral equals the area under the graph, and basically all he does is say that the integral equals area for rectangles, and integrals are approximated by rectangles, so the result carries over for integrals.


This a basic ingredient of apostol's approach, an attempt to convince you that the idea of integral really deserves to be considered area under a graph.

if you do not appreciate this type of argument, it may that apostol is not the book for you. you may still benefit from perusing it, and may profitably do so if it seems enjoyable.

this type of precise characterization of the concept of area, is not even attempted in Stewart and books of that nature. For example, I am looking at Stewart, 2nd edition, section 4.3, where he simply defines "area" to be the integral, for a non negative function.

he also fails to prove that the integral exists for a continuous function, or even fort a monotonic function.

stewart states in thm 4.11 that the proof of integrability of continuous, and monotonic functions, is given in courses in advanced calculus.

apostol proves both of these results, so in stewarts own words, what apostol is teaching you is an advanced calculus course, at least in comparison to stewarts version.

In fact apostol reveal's by his easy proof that the argument showing monotonic functions are integrable is trivially easy, and belongs in every elementary calculus course. but it is not in most of them. (In fact this proof is due to Newton.)

this is the difference between stewart's calculus and apostol's calculus.

you are just not getting more than the tip of the iceberg from books like stewart.

apostol takes you deeper, much deeper.

 

[courtrigrad]

does courant go in with the same depth?

thanks

 

[devious_]

I know that Apostol goes much deeper than Stewart, and I really appreciate that; what I was saying is that I don't enjoy proving things that appear to not require any proof.

 

[mathwonk]

courant does not go into things quite as precisely as apostol. courant is deep, but there is a little more taken for granted. i think courant will assume basic facts from trig as needed, whereas apostol will lay out exactly which proeprties of sina nd cos are going to be used.

they are basically the same properties, but apostol has a fanatical commitment to really spelling out every fact and assumption with complete precision and care.

you can get the idea by going to the library and perusing these two boks, and then comparing with a book like stewart or anton, something along those lines.

apostol and courant are answering questions that do not even arise in books like stewart et al.

 

[devious_]

I think Courant may be the book I'm looking for! I'm going to try to check it out as soon as I can.