I used Stewart's Calculus at that time and I don't have it here lying around at the moment. I don't remember the exact number of the exercise that I submitted as a bug - it may be still in the database if old records are kept. There was some additional stuff connected to it - such as wrong calculated limits etc. As far as I recall, there was some discussion about it then, but no satisfactory solution was suggested. It is nice to see that Maple can be used to find the correct answers in the Wikipedia example. Still it looks like trial and error method - try this way, if it works -OK, if it doesn't - then try this trick, and if it works, but gives a wrong answer, try another trick until you find the way that gives the correct answers (if you know the correct answers.)  If somebody has Stewart's Calculus handy, he or she could post some exercises from it to try. I might do that later - I have to leave now.

Alec 

Yes, I also feel similar. I didn't mean the lack of education - just different direction of it. For example, for a mathematician, the differential equation course is mostly theorems of existence and only a very little bit of practical algorithms for solving some differential equations in very simple situations - usually without even mentioning such things as Laplace transform. For a physicist, the existence theorems don't have any sense - the differential equations that they have, come from real life examples, and there is absolutely no doubt that the solutions exist. Accordingly, the differential equation courses for physicists don't have the existence theorems at all, or only statements of them, without a proof, and are concentrated on practical methods of solving differential equations in many complicated situations, and definitely with heavy use of Laplace transforms.

For Clairaut's theorem, there was a practical difficulty of using Maple in my Calculus III course to calculate the partial derivatives in standard counterexamples. The examples are simple rational functions, with quadratic, maybe, polynomials in the numerators and denominators - nothing especially unusual. And students had to do all the calculations manually because Maple gave wrong answers for partial derivatives.

Alec

PS Physicists, maybe, a small minority, but they do a lot of important work, and I am more familiar with some of them than with some other people working at Maplesoft -Alec

PPS Axiom has some very talented people working on it, and has series of very useful packages and functions, especially in combinatorics. Recently, however, SAGE seems to become the leader in CAS development - certainly, in topics related to Number Theory, but they do a lot to cover current lacunes in other directions, and include a lot of other good software, including Axiom -Alec.

Did you try Google translator? Here is what I get: 

The Twelve-Eleven highlights the left hand:
Because it proposes midnight in the country.
It listens to the pond with offnem mouth
Very quietly cry of the canyon's dog.
The Dommel reckt on the tube
The moss frog from his lying bog.
The Schneck listens to in his house
Likewise, the mouse potato.
The Irrlicht itself makes maintenance and rest
on a windgebrochnen Branch
Sophie, the Maid, has a face:
The moon sheep goes to the high court.
The gallows wehn brothers in the wind.
In the remote village screams a child.
Two Maulwürf kiss to Stund
Neuvermählte than in the mouth.
However, deep in the dark forest
a night Mahr his fists ballt:
Dieweil a late first fuselage wall
not occurred in the pond and marsh.
The Raven Ralf calls scary: "Kra!
The end is here! The end is here! "
The Twelve-Eleven lowers the left hand:
And again sleeps the entire country.
The problem
The Twelve-Eleven came to his problem
and said: "My name is uncomfortable.
As I went about three-four
instead of seven - God pardon me! "
And lo and behold, the twelve-Eleven is called
from that day from Twenty-three.

Regarding x=x+1 - yes, that looks scary. Personally, I prefer x+=1. That didn't work in Maple though if I remember correctly.

No comments about the 12th letter in the alphabet.

So 12 would mean AB. Well, B I understand, but what A might stand for? Certainly not me.

I wonder why it's 18. Seems to be 12 letters. Maybe I spell it wrong.

Alec

What I meant to say is that the same things produce different emotional reactions in different people. And it seems as if some of these reactions may be explained by the educational differences. To become a professional (pure) mathematician, one has to go through a lot of hard work and it takes many years. What am I saying - Axel, having a mathematical education, you know that as well as I do.

Other professionals that I mentioned, physicists, applied mathematicians, and computational mathematicians, also have to go through a lot of things in their education - just through different things. And the math courses that they have are more lightweight so to speak and concentrated mostly on the typical cases and not on the exceptions as the math courses for the mathematicians do.

Some things also related to teaching experience. When you teach Calculus and Analysis courses every year for 30 years, with the accent on the Fundamental Theorem of Calculus, and then see that Maple's integral of continuous functions can be discontinuous, one might get a heart attack. While for people without such experience that means nothing.

The same with the Clairaut's theorem. As I said, courses for mathematicians are concentrated mostly on exceptions and not on general cases, and exceptions are often more important than general rules. So for a mathematician, it is much more important that the partial derivatives may not commute than that they are commute in typical examples. While for people with other educational track, accenting on general cases, and not on the exceptions, Maple's behaviour seems normal and nothing to worry about, especially if it is mentioned in the help pages.

Alec

When I first noticed that (a long time ago), and read about commuting derivatives in the help page, my hair stood up. Many other people don't care about that, as well as about some other things making my hair stand up - like int(x^n, x) being equal x^(n+1)/(n+1) or any matrix in 0 power being equal 1 (these 2 things may be fixed in the latest Maple releases - who knows...)

When I talked about that with some physicist, he said that it is only pure mathematicians' problem. For physicists, applied mathematicians, and computational mathematicians "partial derivatives commute". It would be hard for them even to believe that they may not. And any counterexamples won't help because they exist only in mathematics and not in real life.

That reminded me something that V.I. Arnold said in one of his books. He noticed in one of physical books that it was said about the derivative that it is a mathematical approximation to the slope of the tangent line. When he talked with the author of that book and told him that it _is_ the slope - not an approimation to it, he said - only from mathematician's point of view. In real life (f(x+t)-f(x))/t has sense only for t not less than, if I recall correctly, 10^(-16), and for that value it gives the real slope, and smaller values don't make physical sense because Newtonian physics doesn't work on such small distances and quantum mechanics should be used there instead (with completely different formulas). So the limit that mathematician's use is only approximation to the real slope. Also, he said, you mathematicians write a lot of other wrong things - for example, that the graph of y=exp(-x^2) doesn't intersect the x-axis while everybody can see that they intersect and not that far from 0, and for x=10 nobody can insert even an atom between them.

My guess is that many people who work on Maple, are either physicists, or applied mathematicians, or computational mathematicians, and that explains many such things in Maple which make hair stand up for pure mathematicians.

Also, regarding the Clairaut's theorem, I was told that commuting derivatives are used in many differential equation packages and without that they will be broken. So what is your choice - to have things done right, or to have working differential equation packages?

Alec 

For Lebesgue integral, that may be not true (in case of distributions). For example, while the Lebesgue integral of the Dirac delta over the line is 1, it is 0 for the line without 0. For Riemannian integral, that's how the change of variable theorem is usually formulated - requiring bijectivity (or inversability in other words, or 1-1 and on) of the transformation on some subsets of V and W - so that the most useful cases, such as changing to polar coordinates be covered. And you are right - in usual Calculus courses the Riemannian measure is not defined. Different textbooks use different approaches. One way to define the subset with Riemannian measure 0 is that it can be covered by measurable domains with measure epsilon for any epsilon>0. In this content, the measure means volume in 3d case, or area in 2d case. In good real analysis courses the Riemannian measure should be covered.

I was talking about the change of variables in Riemannian integrals.

From computational point of view, there is not a big difference between Riemannian and Lebesgue integral - they give the same value for domains (and functions) on which Riemannian integral is defined (leaving aside such things that for Lebesgue integral, say, on the segment from 1 to 2, the orientation of the segment doesn't make any difference - there are no such things as Lebesgue integral for x from 1 to 2 or from 2 to 1 - it is always "from 1 to 2"). The integral in Maple is not Riemannian and not Lebesgue - it gives 1/2 for the integral of Dirac delta from 0 to infinity which doesn't have sense in any mathematical content.

Repeated integrals, however, represent a different mathematical object which can be correctly defined as integration of differential forms. The practical meaning of it is that the change of variables in the integrals of differential forms and in Riemannian or Lebesgue integrals is done differently - the integrand is multiplied by the Jackobian for differential forms and by the absolute value of the Jackobian for Riemannian or Lebesgue integrals.

For example, for the area of unit square given as int int 1 dxdy over the square and represented in Maple as int((int 1, y=0..1) ,x=0..1), if we change x to u=1-x, then the first integral should change to int int 1 dudy over the same square, but the repeated integral should be changed to int((int -1, y=0..1) ,u=1..0).

That, probably, introduces some additional difficulties to the correct implementation of the change of variables.

Alec

That's why I wrote "excluding some subsets of measure 0". The integration domain should be compact (i.e. closed and bounded) for usual Riemannian integrals, and that can not be achieved by some changes of variables (including polar coordinates.) Another exclution is for phi, it should be either 0<phi<=2 pi or 0<=phi<2 pi. That's the main reason why excluding subsets of measure 0 is a usual part of standard integration theorems.

Alec

For some reason, that was posted twice and I couldn't find the "Delete" button or link. Is it possible to delete your own post here (that doesn't have replies)?

Alec

Here is an example. Suppose that the integration domain is given by inequality 1<=x^2+y^2<=4. Changing to polar coordinates requires 2 inequality conditions for bijectivity (excluding subsets of measure 0 here and there), r>=0 and, say, 0<=phi<=2 pi. So the new domain is given by 3 inequalities, r>=0, 0<=phi<=2 pi, and 1<= r^2 <=4 which can be reduced to 2 ranges phi=0..2 pi and r=1..2.

Alec

A humorous thing related to that (for people visiting Maple newsgroup occasionally) is that Vladimir Bondarenko developed some new ways of calculating multidimensional integrals (before becoming a Maple integral bug expert.) Being treated more nicely by the company, Maple could have some multidimensional integrals already implemented with his help.