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The Should I Become a Mathematician? Thread

by mathwonk
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Oct19-12, 11:02 AM
P: 83
Part I

so the messy part of cruse and granberg was something to do with Decartes' method of tangents on parabolas and how you're only suppossed to cut the curve once?
Was the problem with with complicated functions the tangent might be hitting multiple points with higher degree curves or somethings? [at least that's the gist of the complaint on amazon]. Didnt other textbooks use that method, and fall into similar traps? or it was just something that worked for some equations and not higher degree equations where it could be strange or messy.

[watch out it's sin(x)/x]

- Any thoughts on:
a. Calculus for the Practical Man - JE Thompson [1931/fixed up 1948 - and obsessed with rates and flows]
b. Quick Calculus - Kleppner
[wasnt that late 60s or early 70s, i don't remember if it came out before or after Introduction to Mechanics... but i thought it was a great gesture, all you need with high school algebra is my 'other book' to read 'my other book]

i was pretty skeptical of the need help with calculus textbooks but two textbooks after the 80s seem to be quite good

c. The Calculus Lifesaver: All the Tools You Need to Excel at Calculus - Adrian Banner - Princeton 2007 - 752 pages
[Banner's style is informal, engaging and distinctly non-intimidating, and he takes pains to not skip any steps in discussing a problem. Because of its unique approach, The Calculus Lifesaver is a welcome addition to the arsenal of calculus teaching aids. - MAA]
[I used Adrian Banner's The Calculus Lifesaver as the sole textbook for an intensive, three-week summer Calculus I course for high-school students. I chose this book for several reasons, among them its conversational expository style, its wealth of worked-out examples, and its price. This book is designed to supplement any standard calculus textbook, thus my students will be able to use it again when they take later calculus courses. The students in my class came from diverse backgrounds, ranging from those who had already seen much of the material to others who were struggling with basic algebra. They all uniformly praised the book for being one of the clearest mathematics texts they have ever read, and because it reviews the required prerequisite material. The numerous worked-out examples are an ideal supplement to the lectures. The only difficulty in using this book as a primary text is the lack of additional exercises in the text. However, there are so many sites and sources for calculus problems that this was not a problem. I would definitely use this book again. - Steven J. Miller, Brown University]
[some wonder about the lack of reinforcement]
[not the best for clarity]
[not always easy to follow]
[for volumes with shell and disks - far more complicated than main textbooks and still leaves out a lot of explanation]

d. How to Ace Calculus

and then there is
e. Calculus - Elliot Gootman - Barron's Educational Series 1997 - 342 pages
[said to be much better than the dummies book]
[and for some more useful than the how to ace book]
[how they do it - 'Once you master about twenty basic procedures, the rest becomes far more approachable. I recommend this book highly to those frustrated with standard textbooks or simply wishing to understand the basics of how calculus works.']

f. The Humongous Book of Calculus Problems: For People Who Don't Speak Math - W. Michael Kelley - Alpha 2007 - 576 pages
[Kelley does a great job of stripping away the gobbledygook and delivering you the nuts and bolts of calculus ON PAR with the "hardcore texts". There are many of those "hardcore" books, and they just dont teach well. What this author has done is to teach you how to solve the problems as well as the underlying logic.]

two older super-obscure classics that i found fascinating opinions on:

g. Calculus - Fadell - early 60s
[considered one of the neater post Sputnik calculus books]
[The best may be the book by Fadell also written in the early 60's which has some fantastic figures and a very unique treatment of calculus]

h. Differential and integral calculus - James Ronald Fraser Kent
[verbose older calc text]
[I picked up a used copy of this text based on the five star review that was given. I think this book proves that all of first year calculus can be covered in a compact book. It assumes the reader has mastered pre-calc math and does not waste time covering much of the pre-req material. However, this book still packs a maximum density of information given its size.]
[The Book is less than half the size of the prototype modern calc text.]
[The text is very wordy and broken down into compact subsections. At points, I felt the author could have done a better job explaining certain topics wi th less words and a few more equations. The figures are also not as good as in other older texts like Fobes or early editions of Thomas. However, this book is still much better than most of the calc books in print today. All in all a very decent older text that is worth digging up if you are into calculus pedagogy.]



i. Calculus and Pizza: A Cookbook for the Hungry Mind - Clifford A. Pickover - Wiley 2003 - 208 pages
[A must see for 9th and 10th grade high school students]
[one word this book is: Enthusiasm]
[This book is the simple solution to every young student avoiding complications in calculus later in life. I was given this book early on during basic algebra (which I wasn't great at). When I finished reading this book I didn't claim to know calculus: I skimmed the first couple chapters over and over. But, I had an idea of what people meant when they said "Calculus."]
[America's public educational systems lack the rigor that is required by its universities and colleges because students are not getting "very basic" ideas early on. This book is a definitive solution. Reading parts of this book in 9th or 10th grade can give students time to let the fundamental simplicity of calculus percolate, something that cannot be rushed in a semester.]
[Students don't need trigonometry, or advanced algebra. They need insight early on. If you're searching for a calculus book because you're having trouble with it now, do your younger friends a favor and recommend this book. It could mean the difference between success and failure when they transition from Precalculus to calculus. This book should be treated the same way astronomy and science survey books are written to inspire interest in young people. Move over earth, life, and health sciences and make some room for Calculus and Pizza - food for the hungry mind.]
[This book served to demistify the entire basics of the calculus for me. Without it, I'd still be wondering about the derivative, or about limits or integrals. On the other hand, it contains about 5% of the stuff in a real calc book, which is why I'm glad I've got both. Even today I refer back to this when the definitions Swokowski gives me are too obscure to understand.]
[If you have trouble understanding calculus, buy this, not a copy of Schaum's outlines. This will open you up to fundamental concepts, and once you have those down, reading any normal calculus text will be a breeze.]
[A really fun read, and you learn Calculus too]
[From the first couple of pages I felt as though I had been thrown in the deep end of the pool in order to learn how to swim. I was anticipating a more accessible book and I was disappointed. The examples of tomato sauce mold, rocket launched meatballs and giant pepperoni (don't ask) didn't serve to ground calculus in the real world for me. Again, maybe a terrific text for people that already have a grounding in the subject, but hardly as comforting as the title would lead you to believe.]

[Yeah, i'm probably the first person to bring up a book called calculus and pizza, but if it is a book that can teach someone calculus 4 years before most people encounter it in school, that's a good thing]

[I recall some book in a 1970 Edmund Scientic Catalogue that had some package or book [i think it was like a book with extra demo materials like cardboard cutouts or something] and the blurb was about how elementary school children could be taught ideas that are in calculus, and i thought that this pizza book is doing similar stuff, and well books that do this sorta thing are rarer than hen's teeth]


the best newer textbook [yeah another Addison-Wesley book, how creepy is that... as i said they always put out good stuff]

j. Multivariable Calculus - William L. Briggs and Lyle Cochran - Addison Wesley 2010 - 656 pages
[used at UCLA]
[Most readable calculus book I've yet to come across]
[I was re-taking multivariable calculus this past semester (as kind of a filler class at the community college. I just had some general ed. class to take, so I thought I'd try calc III again and see if I would actually learn anything about vector calculus this time around). We were loaned out the paperback Multivariable edition of the Briggs/Cochran calculus book. One down-side of these copies - the ink smudged way too easily. But that's really not a factor in my four-star rating, I promise. ]
[I've managed to take long enough getting through school (as I mostly just take evening and online classes, what with working during the day) that I've used three different calculus books - Stewart, Thomas and now Briggs. Also, a friend and I are kinda math/physics junkies so we both have fairly extensive collections of Dover books and other various textbooks. Point being, I've come across a lot of different calculus books.]
[And this one has just become my favorite. It never feels dumbed-down (like Stewart did), and it's significantly more readable than Thomas calculus (which does Ok at times, then falls apart at other times). If you've happened to used the Knight physics textbook recently, the Briggs/Cochran book is similar in flavor - conversational yet extremely thorough. It still requires focused reading and plenty of practice, but at least the book won't be an obstacle to learning - as is the case with so many other textbooks in the math/physics world, I find.]
[Drawing on their decades of teaching experience, William Briggs and Lyle Cochran have created a calculus text that carries the teacher’s voice beyond the classroom. That voice–evident in the narrative, the figures, and the questions interspersed in the narrative–is a master teacher leading readers to deeper levels of understanding. The authors appeal to readers’ geometric intuition to introduce fundamental concepts and lay the foundation for the more rigorous development that follows. Comprehensive exercise sets have received praise for their creativity, quality, and scope.]

[Though I was a little skeptical about a first edition, my skepticism faded quickly after reading through the beginning of the book, particularly limits. Very, very good explanations and examples that thoroughly prepare the reader for the upcoming exercises. The definitions are great, and the graphics are very well laid out and explained. All in all, though I haven't read through the etire book yet, I have read enough Calculus books to know a good one from a bad one. This being a very good one.]
[Will never be as popular as Stewart's Calculus, and it is far from being a serious, self-respecting Calculus book - such as the one written by Apostol. Not a good text-book for students in Science and Engineering who need to have a better understanding of Calculus and applications, based on more serious Engineering and Physics-born examples, with more serious computations and proofs!]
[This book is actually pretty good, good for self study. But if you want a really good book, I would recommend Ron Larson's Calculus book instead.]
[The book would be great for a high school student who is trying out Calculus, but is not good at Math at all. It may be good for the Liberal Art student pursuing multidisciplinary studies: that is, a mixed salad of Humanities, Education, Social Sciences and Life Sciences, spiced up with some Calculus just for the sake of sounding like a true intellectual!]

[it's got some moody blue and black artwork on it too]


k. Jerrold E. Marsden and Alan Weinstein - Calculus I, II, III - Springer-Verlag [came out in the late 70s or early 80s] second edition is 1985 is all i know about it.. and it was used at Berkeley, since i think Marsden is there and cranks out 3-4 textbooks through the decades...

I heard extremely little about it, any ideas on when it first came out, and how the different editions are, by anyone out there?
Im sure people didnt like 3 orangey yellow textbooks with 3 study guides and then you possibly get pushed into marsden's vector calculus textbook afterwards...

heck here is a neglected text these days from the 60s

l. What Is Calculus About? (New Mathematical Library) - W. W. Sawyer
[someone should talk about one of the first NML books, i thought they were one of the greatest ideas around, a huge series of books to supplement you from high school on up]
[i think the closest anyone came to something sorta like that might be the oxford chemistry series that had all these strange silver and back thin 80s paperbacks which were like 50-70 titles i think...]

[physics only had the anchor science series for teenagers, and man those arent easy to see, but you could always see a few in the bookstores of the 70s, usually the electronics book or some of the history books] It looked like so much promise in the 60s and it petered out in the 70s with the PSSC texts [or likely nixon gutting the libraries and education funding stuff that got pushed 1960-1968]


another newish one that looks good

m. Calculus: The Elements - Michael Comenetz
[Best Textbook on Calculus - Concise & Fun to Read & Comprehensive]
[It's no doubt that Stewart's book is the most popular textbook on Calculus. It's comprehensive and standard. However, it's a pain to read through every page and do all the exercises.]
[In that regard, I've found Michael Comenetz 'Calculus: the Elements' most suitable for students without a solid background who intend to major in physics, math, chemistry, and engineering. Comenetz' book is not only comprehensive but very concisely written. Problems are well chosen - unlike Stewart's that has repetitive/similar problems all over the textbook. Yet, my advice would be 'keep Stewart's as a reference while learn from Comenetz's' This, based on my own experience, is the most effective to achieve high scores in tests and excellent grades.]
Oct19-12, 11:03 AM
P: 83
Part II

off topic but a 'friendly' book as in the rudin path to math texts

n. Advanced Calculus: A Friendly Approach - Witold A.J. Kosmala - Prentice-Hall - 700 pages - 1998
[I have copies of Rudin, Apostol, Bear, Fulks, and Protter, but this book beats them all as an introductory text. If you are looking for a self-study text, or if you want a reference companion to help you understand Rudin or Apostol, try this book first. You won't be disappointed.]
[The author of this book has used " a friendly approach " to present the stuff so that readers will actively be engaged in learning with less strain. This has not in a sense simplified the difficult elements of Calculus but bringing along the readers to think and reason while studying the subject.]
[Designed to be readable and intimidation-free, this advanced calculus book presents material that flows logically allowing readers to grasp concepts and proofs. Providing in-depth discussion of topics, the book also features common errors to encourage caution and easy recall of errors. It also presents many proofs in great detail and those which should not provide difficulty are either short or simply outlined. Throughout the book, there are a number of important and useful features, such as cross-referenced functions, expressions, and ideas; footnotes which place mathematical development in historical perspective; an index of symbols; and definitions and theorems which are clearly stated and well marked. An important reference for every professional who uses advanced math.]

For the last huff, jump in anyone....

o. Calculus With Analytic Geometry - 9th edition 2008 now....
[Ron Larson and Edwards] or [Larson, Hosteller and Edwards] - DC Heath and Brooks/Cole

people think the highest and lowest of this textbook, though it's been through a hell of a lot of editions, and i think in the 80s it flaked out with some computer gunk and then went back to basics....

the comments are all over the place *grin*

[this isnt Edwards and Penney]
[liked by Alexander Shaumyan - New Haven, CT]
[easy to follow]
[it doesnt really explain things adequately]
[it skips too many steps in the examples]
[some think it's got a nice format and easy to follow]
[too software fixated with frills and fluff and fad though]
[Excellent treatise of 3-semester calculus. A classic]
[Decent text but by no means excellent - 3 out of 5 rating]
[if people complain this book makes calculus too simple, so what? If you are struggling and can't do the easy stuff, then how on earth are you going to start doing the hard stuff later on?]
[i get the feeling this book isnt better than Sherman Stein's or Thomas and Finney really]
[starts off simple, but then goes into too many shallow applications, with skimpy second year stuff]
[I have many of the same criticisms of this book as I do of the Stewart, although I do think this book does a slightly better job in the very beginning, for example, when introducing the limit, and also in that it leaves out some of the extraneous and confusing attempts at applications in the first chapter. I still think the book contains too many confusing applications from the second chapter onward. I do think the book would be improved by having a completely separate section covering the definition of the limit, however.]
[I like the prose in the examples. I like the presentation of some of the material from multivariable calculus. But again, this book is like a typical intro calc book - it's not rigorous enough, has too much brute force, too many applications, not enough mathematics, not enough creativity. This book doesn't cultivate the awe and wonder that should be present when a student learns calculus.]
[There is no text, in my opinion, more suited towards use in any introductory Calculus series, but this text is also ideal for self-study. The theory is presented in crystal clear fashion, and then multiple examples are given in order of increasing complexity.]
[just another junk book]
[This book does provide the concepts and theory critical to an understanding of calculus. Unfortunately, it is in a wordy, technical, abstract, and thoroughly annoying format. I used this book for calculus 1 and 2. However, unlike my classmates, I learned all the material from an engineering math book (kenneth stroud, engineering mathematics).This book gives you plenty of abstract proofs that look like bull@!#t, but falls far short of my engineering book in encouraging an understanding of calculus. The truth is, this book gives you hundreds of formulas to memorize, instead of a relative few like my engineering book that can cover every problem. Most importantly, I can create these formulas if I need to, because I actually UNDERSTAND what is going on. By the way, I got an A+ in both courses, and I never bothered to learn the epsilon delta crap.]

i ain't got much of a timeline on the book but i got this much
[First Edition]
[Second Edition]
[Third Edition] [started to use computer generated graphs - ugh]
[Fourth Edition] 1993 [started to use computers and graphing calculators - ugh]
[Fifth Edition] [started to use a CD Rom - ugh]
[Sixth Edition] 1998 - 1316 pages [started to do stuff online - ugh]
[Seventh Edition]
[Eighth Edition] 2005 - 1328 pages - Brooks/Cole
[Ninth Edition] 2009 - 1328 pages - now just Larson and Edwards

oh one more

p1 and p2. Lang's simple and non scary calculus text, came out in like 1964 for a basic course, and through the changes in curriculum people found that it's still useful today...

p1. A First Course in Calculus - First edition - Lang - Springer 1964 - 264 pages
[reissued in the past decade as - Short Calculus - yeah the first edition is back]

p2. A First Course in Calculus - fifth edition - Lang - Springer 1998 -752 pages
[the bloated new editions]

the comments:
[simple, but not unsophisticated]
[As a high school teacher, I used this text with great success several times for both AP Calculus BC and AP Calculus AB courses. It is my favorite calculus text to teach from, because it is very user-friendly and the material is presented in such an eloquent way. There are no gratuitous color pictures of people parachuting out of airplanes here. Opening this book is like entering a temple: all is quiet and serene. Epsilon-delta is banished to an appendix, where (in my opinion) it belongs, but all of the proofs are there, and they're presented in a simple (but not unsophisticated) way, with a minimum of unnecessary jargon or obtuse notation. He doesn't belabor the concept of "limit"; most calculus books beat this intuitively obvious concept into the ground. Even though it doesn't cover all of the topics on the AP syllabus, I would rather supplement and use this text rather than any other. - B. Jacobs]
[Calculus for beginning college students]
[I needed to bring my high school calculus up to speed for first year physics studies and found this to be the only book which covered the necessary ground. The material is presented in a thorough manner with the great majority of topics shown with proofs. The book is very well organized and there are abundant worked examples. Some problems are offered which deal with matters not covered in the text, but usually there is a worked example given among the answers. Lang deals with the material in a clear fashion so that the subject matter is usually not difficult to follow.]
[On the negative side I can say that there is no human touch between the covers. His sole attempt at humor is an item following a list of problems in which he notes "relax". In the foreword he exhibits his firm belief that many freshmen arrive unprepared for college calculus, which may be true. But nowhere in the book is there a note of encouragement, so it cannot be described as reader friendly. Finally the index is pathetic - just three pages for a book of 624 pages, so that finding things can be frustrating.]
[Effectively conveys key concepts and skills]
[Serge Lang's text does an effective job of teaching you the skills you need to solve challenging calculus problems, while teaching you to think mathematically. The text is principally concerned with how to solve calculus problems. Key concepts are explained clearly. Methods of solution are effectively demonstrated through examples. The challenging exercises reinforce the concepts, while enabling you to develop the skills required to solve hard problems. Answers to the majority of exercises (not just the odd-numbered ones) are provided in a hundred page appendix, making this text suitable for self-study. In some sections, such as related rates and max-min problems, Lang provides many fully worked out solutions.]
[As effectively as Lang conveys the key concepts and teaches you how to solve problems, he does not neglect the subject's logical development. Topics are introduced only after their logical foundations have been laid. Results are derived. Theorems are proved when Lang feels that they will add to the reader's understanding. Through his exposition and his grouping of logically related exercises, Lang teaches the reader how a mathematician thinks about the subject.]
[The book is divided into five sections: review of basic material, differentiation and elementary functions, integration, Taylor's formula and series, and functions of several variables. The heart of the course is the middle three sections.]
[Most of the topics covered in the review of basic material should be familiar to most readers. However, it is still worth reading since there are challenging problems, properties of the absolute value function are derived from defining the absolute of a number as the square root of the square of the number, conic sections and dilations may be unfamiliar to some readers, and Lang views the material through the prism of a mathematician who knows what concepts are important for understanding higher mathematics.]
[Lang introduces the derivative as the slope of a curve in order to motivate the introduction of the idea of a limit. Next, Lang teaches you techniques of differentiation and shows you how to use them solve applications such as related rate problems. After a detailed discussion of the sine and cosine functions, Lang introduces the Mean Value Theorem and illustrates how it can be used for curve sketching and solving for maxima or minima. Lang covers properties of inverse functions before concluding the section by defining the natural logarithm of x as the area under the curve y = 1/x between 1 and x and defining the exponential function f(x) = e^x as its inverse.]
[The integral is introduced as the area under a curve, with the natural logarithm taken as the motivating example. Lang explains the relationship between integration and differentiation before introducing techniques of integration and their applications. Integration with respect to polar and parametric coordinates is introduced to expand the range of applications. The exercises introduce additional tricks that enable you to solve integrals that do not succumb to the basic techniques. A table of integrals is included on the inside of the book's front and back covers.]
[Lang's demonstrates the power of differential and integral calculus through his discussion of approximation of functions through their Taylor polynomials. This chapter should also give you an idea of how your calculator calculates square roots and the values of trigonometric, exponential, and logarithmic functions. The behavior of series, including convergence and divergence tests, concludes the material on single variable calculus.]
[The material on functions of several variables in the final section of the book is covered in somewhat greater detail in Lang's Calculus of Several Variables (Undergraduate Texts in Mathematics). Since the corresponding chapters in that text include additional sections on the cross product, repeated partial derivatives, and further techniques in partial differentiation and an expanded section on functions depending only on their distance from the origin, I chose to read these chapters in Lang's multi-variable calculus text. The material that is included here, on vectors, differentiation of vectors, and partial differentiation, should give the reader a solid foundation for a course in multi-variable calculus.]
[I have some caveats. There are numerous errors, including some in the answer key. Some terminology is nonstandard, notably the use of bending up (down) for concave up (down), or missing, limiting the text's usefulness as a reference. In the chapter on Taylor polynomials, when Lang requests an answer accurate to n decimal places, what he really means is that the error in the answer should be less than 1/10^n, which is not the same thing. At one point, Lang claims that the Extreme Value Theorem, which he leaves unnamed, is obvious. I turned to the more rigorous texts Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition) by Tom M. Apostol and Calculus by Michael Spivak, where I discovered proofs covering one and half pages of text of the Extreme Value Theorem and a preliminary result on which it depends that Lang does not state until an appendix much later in the book. Perhaps Lang meant the Extreme Value Theorem is intuitive. While I found much of the text to be clear, I sometimes found myself turning to Apostol's text for clarification when I read Lang's proofs.]
[Despite my reservations, I think this text is well worth reading. Reading the text and working through the exercises gives you a good understanding of the key concepts and techniques in calculus, enables you to develop strong problem solving skills, prepares you well for more advanced mathematics courses, and gives you a sense of how mathematicians think about the subject. ]
[a book that focuses on the foundation without trying to do too much and it does that very well. self-contained and easy-to-follow, this book promotes understanding of the basics]

mostly recent stuff i packrat into my books for calculus... but i figure that almost any of these books should be worth discussing here, by anyone who's got a copy, used a copy, browsed it in the library, or utterly hates the book.....


another thing to talk about, what were the MOST popular textbooks out there 50s or earlier to today?

Thomas and Finney seemed popular [i wonder if thats because it was just enough to make engineering happy, as well as the math majors and the people who just need calculus once]
[I heard the alt editions were better, and what were those, it sounded like all the unreadable fluff and proofs were yanked out, but those only came out in the 60s or 70s, and the alt editions i think had unique numbering]

and i do recall

[i also think the writing of the 9th edition is actually clearer than in thomas original book - mathwonk]

I'm not sure of story, but wasnt the second edition pushed out really quickly for thomas, and i'm wondering if the first edition had problems, or just so much more was written but not fully completed for the first edition, and well, when the book took off, he said, i finally finished the last few chapters which i needed a few more years to finish up.... etc etc

stewart i think started to get popular about 1990 or so..

what was always surreal is how some older bookstores would just carry stacks and stacks of the 1967-1974 textbooks for calculus, which were all the mainstream, dont take too many chances, write for all audiences, and keep all that formalism, don't make the book too easy, don't make it too eccentric, don't stick in any material if the other top 7 sellers don't include it..... and no one would buy them at 10 dollars and you'd see 15 years of dust on them....yet they would be great books for 2 dollars for the store to dump on people who want 'supplementary reading'

i always thought that the super easy books were far better, and the super difficult ones... the books in the middle just were compromised far too much, and lacked any vision...

any why is that no syllabus around tosses a schaum's outline for calculus or physics on the list?
Oct19-12, 11:29 AM
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my perception of a calculus book is partly influenced by when it came out relative to my math education. kleppner i believe was a harvard physics prof who wrote his book after i had taken a spivak style course from john tate at harvard (as spivak also may have), so did not interest me at the time.

gootman is one of my favorite books for struggling students and i have a copy signed by gootman, my long time colleague and a master teacher and analyst.

i liked lang's calculus books and learned how easy and simple riemann integration is from them.

i loved comenetz's book, and wrote the initial rave review of that book. unfortunately i gave away my review copy as a prize to a good student. I attach my (edited) review, no longer available on the publisher's website: (see below)
unfortunately for the buyer, the price has increased from under $40 to over $125. Perhaps that is one reason my review has been removed, since it originally contained a grateful comment about the price.

i loved the first edition of edwards and penney, two wonderful scholars and teachers and friends of mine, but to my taste the book did not improve through several editions apparently designed to enlarge its audience at the behest of the publisher. it seemed to serve as the model for stewart's book.

schaum's outline series was wonderful in the old days, extensive and good problems, plus brief and useful theoretical summaries; but more recently when i tried to use it in a course, it seemed greatly reduced in quality and usefulness somehow, no longer worth it.

the elementary error in cruse and granberg is the fact that the fermat criterion for a tangent line is not that the polynomial which vanishes where the line meets the curve should have only one root, but that it should have a double root at the given point.

this is easy to check for polynomials where one can always divide (x-a) out from
[f(x)-f(a)]/(x-a) because of the first forced root, and after doing so, simply set x=a to see if there is a second root. the result is as usual that the slope of the tangent line to y = x^n at x=a is na^(n-1).

in fact i have experimented using this method to teach derivatives to undergrads, for polynomials. of course more analysis is required for transcendental functions like sin, e^x.

i wrote out this result in complete detail for the author and publishers when they commissioned me to review the book prior to publication, but they ignored it. perhaps the authors did not understand it either, but i suspected at the time, the book was already ready to go to print and thy did not care to know its flaws.

i have written this method up completely with examples in the class notes attached to post #6 of this thread:
Attached Files
File Type: pdf comenetz review 2 edited.pdf (78.7 KB, 15 views)
Oct19-12, 11:04 PM
P: 239
In reponse to RJinkies partI above:

My calculus course is using Briggs this semester. I think it is a pretty good book... but I feel the exercises are too easy. The explanations are good, though. Definitely better than what I've read of Stewart. Actually, my favorite "popular" calc book is Thomas, I think. There are tons of exercises (100+ per section typically); some of which I've found are also in Apostol and Spivak (decent selection of proof problems). However, certainly not as good as Apostol, Spivak, Courant...

I read a while ago a suggestion for Calculus by Kitchen (forget first name) from mathwonk... I happened to see it in a university library today. Looked like a nice book that covers a lot of material most other books do not.
Oct20-12, 10:28 AM
P: 18
Hi, I wanted to introduce myself. :)

I have recently discovered that math is my calling, and am studying it at a small 2-year college before transferring out next Fall to pursue my BS. I'm taking Calc 1 right now with a Stewart textbook (though due to the earnest recommendations for it all over this site I have ordered Spivak's Calculus as well) and am doing well, though there is a definite change in difficulty level between Pre-Calculus math and Calculus. It's actually quite exciting to me because I remember finding myself so bored in other classes when I could easily predict where my teachers were going with every idea, and the course I am in now is a lot closer to my pace.

Out of curiosity, does anyone know what the best colleges/universities in Florida are for a solid math education? I live nearby UCF so it is my most likely option, but I want to consider others so as to avoid my grad school speaking at me in a new language. And I've heard of a lot of people having issues with UCF's massive enrollment, primarily that of never getting a chance to connect with your professors.

Secondly, I've looked at a lot of grad school programs and they recommend acquiring reading fluency of mathematical texts in French, German or Russian. Which one(s) are most useful to learn, in your experience?
Oct20-12, 11:28 AM
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I "read" French, German, and Russian, well enough to pass a grad school math proficiency test, but only French well enough to actually read a math paper fairly easily.

As far as Russian goes, so few English speakers read it that most big Russian journals are routinely translated into English.

I staggered through a few sections of Riemann's papers in German but even those are at last available in English.

I always thought I could read Serre's clear papers in French, but boy the English version of Algebraic groups and class fields is much easier to get something out of.

So while it is recommended to learn these languages, at least french, and less so german, most of us get by quite well in english, occasionally having to struggle through an original language with a dictionary. but even to do that you need to know the basics of the language.

i.e. learn what languages you can, but be aware that you will be able to read almost everything written fairly recently in english. original languages are needed especially for reading some important works from the 19th century and early 20th cent.

e.g. with my weak german, i still have not read the great paper on linear series on algebraic curves, treated purely algebraically, including an early algebraic proof of the riemann roch theorem, by brill and noether.

it was kind of entertaining trying to struggle through a russian textbook on vector spaces (vyektornye prostranstva) when i kept running across the same words (ochevidno shto and silno) over and over, which turned out to mean "obviously" and "clearly"!
Nov4-12, 08:38 PM
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you described your career progress a few times, but not remembering exactly, could you tell us: Did you study anything (Mathematics) while you were a meat-lugger, not in school? Or did you just work your labor job without studying your subject?
Nov4-12, 10:58 PM
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thats a little like asking country joe mcdonald what he remembers about the 60's, and he answers "nothing".

those were stressful days. suffice it to say i did very little math then. my progress went backwards for a while. i would say i lost about 10 years of my math career in there somewhere.

this is not a thread for discussing politics, but that was a great distraction. those were years when we were fighting fruitlessly in vietnam, nixon was president and using the resources of the government for spying on his "enemies", mayor daley's cops in chicago beat peaceful demonstrators as well as news reporters at the democratic convention, my friends were dying in the war, police and covert government agents were harassing demonstrators, tapping phones, arresting people, hundreds of thousands of citizens were marching in washington dc, there were demonstrations for grape pickers and other farm workers. mlk jr and bobby kennedy were assassinated. it was hard to focus on just preparing for a narrow scientific career. so although i occasionally still tried to read some pages of grothendieck, at age 27 i knew significantly less math than i did at 24 or 25. the one advance i made in those years was by assisting/grading in honors calculus, i had to read spivak's calculus book, and learned a lot of calc i should have known much earlier. reading grothendieck was of no help at all, sort of like telling an average 10 year old he needs to read dieudonne'. you learn by reading important basic material in a good concrete treatment and computing examples, not by reading fancy abstract stuff that is over your head. i also graded algebraic topology and learned some of that, as well as the new york times every day for a while. i still remember reading in 1970 that ARCO oil company had paid zero income tax for several years running on a profit of over 300 million dollars, which was a lot in those days, because of favorable tax codes that allowed oil companies to get a credit for the fact that they are gradually depleting the publicly owned oil supply, called the "oil depletion allowance". i.e. they got a tax break to compensate them for the fact that their free ride was slowly running out because of their own exploitation of public resources. this may still be in place. If we want to discuss these issues we should take it elsewhere though. but these issues distracted me from math for a few years. [/QUOTE]
Nov10-12, 03:46 PM
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has here been any older mathematicians (30+) who've made any impact on mathematics (if so who)? Reading up on mathematicians it seems as though everyone makes great work in their early twenties then just die down
Nov10-12, 03:47 PM
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Quote Quote by converting1 View Post
has here been any older mathematicians (30+) who've made any impact on mathematics (if so who)? Reading up on mathematicians it seems as though everyone makes great work in their early twenties then just die down
Nov10-12, 04:12 PM
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Guess it's never too late then! Thought I had little time left seeing as I'm 17,

Nov10-12, 07:58 PM
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has here been any older mathematicians (30+) who've made any impact on mathematics (if so who)?
30 isn't that old. Actually, very few mathematicians today even get to the point where they can make any significant contributions UNTIL they are about that age. The average PhD age is like 27 or 28, and my impression is that postdocs were this extra thing that they had to stick in because a PhD isn't really enough to become a mathematician anymore. So, by the time you are done just getting started, you're that old.
Nov10-12, 08:39 PM
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30 isn't that old to start or to finish. A lot of mathematicians ``made impact" beyond their 30's. Andrew Wiles, for instance, missed the Fields Medal by a few months.

As a general rule though, don't think about making an impact. Every mathematician who's active and writing papers is changing mathematics, of course, to different extents. To paraphrase Robion Kirby, don't worry about the significance of your mathematical results, worry about being the best mathematician you can be, and the rest will follow.
Nov11-12, 12:53 AM
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Take a look at George Polya, who started late relative to a lot of others (consider also that mathematics has exploded since 100 years ago) and didn't start studying mathematics:

Born in 1987, got the doctorate in 1912 so got the doctorate at the age of 25 (but please put that into context for mathematics especially probability at that time, and I am not denigrating Polya when I make these statements).
Nov11-12, 08:51 PM
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whether or not you will do important math is not determined by your age, surname, gender, or anything else. It is based on your desire. go for it.
Nov12-12, 09:20 AM
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Just turned 36. Still an undergrad. Not giving up. :)
Nov27-12, 04:51 PM
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regardless of what I do re: staying in NYC vs. Brandeis program, I'm gonna take some math in the spring semester. Seems like it makes sense, for continuity's sake, to take real analysis II.

Was also thinking Algebra I. Thoughts?
Nov30-12, 01:34 PM
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Well, I have read some of the posts about textbook recommendations and want to offer an insight of my own:
Normal calculus textbooks? Don't bother. Don't read them, they do more damage, than good. The best thing to do is pick up a Russian Analysis textbook, like Fihtengolz, Zorich or Kydriatsev. They all come in 3 volumes.
Also no textbook is good without exercises. For this the best one by far is Demidoviche's "A Collection of Problems in Analysis".
The other essential thing for mathematics is linear algebra and analytic geometry. Serge Lange has very good book in linear algebra.
But the most important thing is not just studying at a university. You should look for open seminars. These seminars will give you much greater knowledge, than any course ever would.

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