Cours d'Analyse Mathématique: Problems Too Challenging?

[mruncleramos]

Has anyone read Cours d'analyse mathématique (Course in Mathematical Analysis)? If you have, do you find some the the problems just a might challenging? I've found that this is pretty characteristic of books of that time period. Why have problems in textbooks today gotten so much easier?

 

[arildno]

I don't know Goursat's work, but:
Textbooks today are in general "dumbed down" compared to the textbooks of earlier days.
This is as true of maths books and physics textbooks (and newspapers for that matter, but I'm beginning to rant here).

 

[mathwonk]

more people go to school than before. maybe that is it. a friend of mine did a vertical study of high school algebra books from 1890 to 1990 and the differtence in problems was amazing, the problems in the oldest books were essentially undoable by anyone today. i am acollege profeesor and i was really challenged to do just some of the algebraic simplifications he showed me. I had to use knowledge no high school student was likely to have to do them.

even standard current books like edwards and penney calculus. 6th edition, are enormously dumbed down even from their own first edition a few years ago.

education is a business, and the customers are parents and students who do not want education, but want high grades. that is not the truth of course for everyone, or even most people on this forum, who are mostly in the minority of interested learners.

first editions contain the author's vision of what the book should contain. if successful, later editions are dumbed down to sell more copies. always buy the first edition of a math book to get the best version of the material.

 

[matt grime]

Goursat is a magnificent book. I occasionally use it to show my students what they "ought to be reading". Never fails to scare them. You may wish to bear in mind that "first courses in calculus" are often intended not just for mathematicians but for engineering undergraduates who have different needs of the subject.

 

[shmoe]

even standard current books like edwards and penney calculus. 6th edition, are enormously dumbed down even from their own first edition a few years ago.

In my first calculus class, my classmates used e&p 4th edition while I used a (free) 2nd edition. I didn't notice a major change in the content, but what struck me was the gratuitous use of colour and pretty graphics to help those limited attention spans. more colour=>more copies sold? I don't know how the 1st and 2nd editions compare here, I've never seen a 1st.

 

[fourier jr]

Has anyone read Cours d'analyse mathématique (Course in Mathematical Analysis)? If you have, do you find some the the problems just a might challenging? I've found that this is pretty characteristic of books of that time period. Why have problems in textbooks today gotten so much easier?

hardy's 'a course of pure mathematics' is similar. he's got some evil problems in there, and unlike many other books, he says where they came from too.

 

[mruncleramos]

My library limit for Goursat's book(s) is dwindling down rapidly. Does anyone know where I can get a new copy? I hear dover had them before.

 

[mathwonk]

E&P over the years has evolved from an exciting and challenging book tightly organized, with no precalculus background, into a tiresome and hugely overweight one with numerous extra chapters on precalculus material, and large numbers of very easy, almost trivial problems. The original book had only challenging problems.

Rearranging of topics also introduced errors such as forgetting to prove the sum law for integrals after moving the topic of antidifferentiation the proof had been based on, or omitting chapter headings in some editions for topics like implicit differentiation.

Other changes include omitting the more significant of the three laws of Kepler which were originally the main application of several variable calculus.

Other changes include simply dumbing down the alngauge used, smaller words, shorter sentences, as well as the annoyingly distrcting increased use of color for the video game generation.


The changes in the problem sets make it harder to assign problems. One used to simply say do the odd problems 1-20, say, but now that will result in doinjg 10 identical and trivial problems, all about parabolas.

For years the proof of the fundamental theorem of calculus had a gap in it, from ignoring the behavior of the function at the interval endpoints, but that was finally filled in later editions.

The proof of the chain rule in early editions also gave the false impression introduced unfortunately by Hardy in his excellent book, that the traditional proof of the chain rule is lacunary, whereas the lacuna is easily filled, as was done in many good books before Hardy's.

E&P 1st edition began with a list of useful interesting applications of calculus, then plunged right into the subject and began deriving those applications. Problems were challenging and deep applications were included at the appropriate places. Later editions delayed the introduction of calculus, introduced huge numbers of more elementary topics and elementary problems, and deleted the more significant applications, and introduced pointless color illustrations and computer related projects and software, as well as web based problem solutions.

In the early editions one had to draw the pictures for the problems oneself, while in the later ones the pictures are already drawn, rendering the volume problems essentially trivial.

There are also errors in some problems, of the sort found also in the differential forms notes followed recently on this forum, in which the Riemann integrability of some unbounded functions like x^(-1/2) on [0,1], is falsely assumed because the antiderivative is continuous everywhere.

I.e. integrability is confused with antidifferentiability.


Still with all these flaws, E&P is still better than most other commonly used books, just not as good as it was itself before the changes.

The same unfortunate phenomenon befell Thomas, Cooke, and Finney in the 10th edition, and Stewart in the third edition.

 

[mathwonk]

that whole chapter one in the 4th edition of E&P was not there at all in the first edition. look at all the pictures for the problems on pages 368-371 of the 4th edition, or pages 149-154; were they there in the second edition? they were not in the first.

the picture for the originally most difficult volume problem in chapter 6 is even in the table of contents in the 4th edition! All of the first 40 differentiation problems on page 115 are pretty trivial.

proofs of the limit laws, the chain rule, and taylor's theorem have all been moved to the appendix in the 4th edition. I'd guess they were still in the text for the 2nd edition.

look on pages 306-307 of the 4th edition, where they state the fundamental properties of the integral. Due to rearranging topics and omitting proofs, they completely forgot in this edition even to state the addition rule for integrals!

on page 312, line 5 of the statement of the FTC says that for F to be an antiderivative of f it means that F'(x) = f(x) for all x in the open interval (a,b), nothing about continuity of F at a and b. Then the proof of part 2 below uses the corollary 2 of the MVT from page 215, which however is stated incorrectly because part of the hypotheses (the continuity of the antiderivative) is given above corollary 1 on page 214.

The point is that the proof of the FTC is incomplete unless at least lipservice is paid to the need for an antiderivative to be continuous on all of [a,b].

on page iv of the contents they left off the word "Implicit" for the description of section 3.9.

This is the kind of thing that can occur when an originally well written book is revised.

kepler's 3 laws still all survived in the 4th edition, pages 684-689, but not much longer.

nonetheless it is hard to find a better book commonly available out there today, except for the usual suspects: Spivak and Apostol intended for math major super honors courses.

So it is the intelligent ordinary honors students wanting an in between book who is getting shafted today. There is simply almost no book for him/her.

 

[matt grime]

Lip service to continuity conditions... are engineers and physicists even aware that there are non-smooth functions? Half the time you think not, but then they go and shove in a heaviside function of dirac delta (one of which isn't even a function) without blinking...

 

[mathwonk]

well presumably they believe in step functions, or they should keep away from cliffs.

 

[shmoe]

I didn't notice a major change in the content...

I should add the caveat that I'm not very observant

Most of my comparisons between the 2nd and 4th was in the problems, in doing the ones from my text and helping friends with theirs. More often than not there was no actual change at all in the problems, except for the odd number change and added diagrams, students can't draw their own trees and buildings anymore I guess. There is a 6th edition in my office and it comes with a CD that promises animations of geometric examples, even less seems expected of their imaginative powers.

I haven't really looked much at e&p since I took the course. My teaching and ta experience has been from very inferior texts, the exception of Spivak one summer. I'll see if I can track down a 1st ed. of e&p to compare with a 6th.

 

[mathwonk]

the first objection to, and hence the first change in EP, occurring already in the second edition, was that the problems had been too hard. i do not recall when the elementary first background chapter was added.

 

[shmoe]

That just makes me feel cheated. This cycle of decline is frightening. In 10 years will we have a Fractions 101 university course? Actually that might be needed now.

 

[matt grime]

Some of the unlucky PhDs at Penn State with me had to each a class abuot how to do fractions (i think they called them oh courses as in remedial help for people we can't refuse entry cos it's a state university, or if i were doing it "oh woe is me" courses for the sheer mind numbing boredom of it.)

 

[fourier jr]

a fractions course? i thought everybody knew that $$\frac{a}{b} + \frac{c}{d} = \frac{a+c}{b+d}$$ :yuck: