Timeline for Why do we teach calculus students the derivative as a limit?
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| Feb 9, 2011 at 2:30 | comment | added | Misha | @fedja:---How exactly do you suggest to define "local uniformity" to the students?---I know what you are trying to push me into. You want me to say "local means in some, probably small, neighborhood of every point, and then we can get global estimates for any compact subset by using the finite sub-cover property." And then you would say: "Aha! it's the same as epsilon-delta! Aha! You need compactness too!" But this would be at the point of my exposition when the uniform theory is well developed and understood already. So it's not the same. | |
| Feb 6, 2011 at 17:44 | comment | added | Misha | It looks like you take the ideological part of mathematics way too seriously, and I don't. As Michael Atiyah said, "...the axiomatic era has tended to divide mathematics into special branches, each restricted to developing the consequences of a given set of axioms. Now I am not entirely against the axiomatic approach so long as it is regarded as a convenient temporary device to concentrate the mind, but it should not be given too high a status..." More on micromath.wordpress.com where among 2 top posts one is put there at my suggestion, the other is a link to the Rokhlin's lecture. | |
| Feb 6, 2011 at 17:25 | comment | added | Misha | @Fedja:---"Locally nice" is the same as epsilon-delta.--- Not really. You can develop all the uniform theory, and then go interval by interval if you have to. You don't start with "locally," you get to it gradually. By the way, my account is still suspended till 3/2/2011 and I can't edit my answer any more. Four chapters (out of 10) of Karcher's lecture notes are already translated into English, after ch. 5 is done, we'll put it on the web somewhere and you can read them. He's jumping to epsilons and deltas too early to my taste, but you may like it more for that. | |
| Feb 6, 2011 at 17:08 | comment | added | Misha | Also: "The notion of continuity is quite foreign in calculus. Monotone or piecewise monotone functions are much more at home. For them continuity is local surjectivity." And: "A strategic solution would be eliminating the calculus, but it is dangerous socially: mathematicians all would get unemployed.Should be replaced eventually but very cautiously. Meanwhile it may be made a bit more intelligible. What makes definitions of limits and continuity bad is chains of 3 quantifier. There are standard ways of fighting with them. Introduce the notion of neighborhood and one quantifier is out. | |
| Feb 6, 2011 at 16:55 | comment | added | Misha | Have you read the Rokhlin's lecture? I had an interesting e-mail discussion with Oleg Viro who took part inn transcribing the tapes. He said, "The problem with calculus is that it occupies a strategic position in teaching of mathematics being quite ugly. It creates negative, foolish image of mathematicians. I do not know how to resolve the situation. A partial step would be turning calculus into a sort of mathematics where everything can be understood by the real students. In particular, eliminate limits that cannot be understood by a student with insufficient training in logic." | |
| Feb 6, 2011 at 16:45 | comment | added | Misha | @fedja:---That is a great step towards where we can possibly converge. How exactly do you suggest to define "local uniformity" to the students?--- Well, the estimates may deteriorate near singularities and you stay away from them or adopt a more permissive modulus of continuity. In some intervals the constants may be better, and you may take advantage of that. It's just common sense. I am not into writing a Bourbaki-style treatise, it's your cup of tea. I go by problems and examples. BTY, why do you want to converge? | |
| Feb 6, 2011 at 16:23 | comment | added | fedja | ---but the world (and even mathematics) is messy, and you better get used to it--- Yes, the world and even mathematics are messy in places but somehow I belong to the school that prefers to "fight the mess with order", not to "get used to the mess". Probably, I had some childhood problems and my parents and teachers told me a few wrong things that got stuck too well. It is too late to try to change that now :). | |
| Feb 6, 2011 at 16:15 | comment | added | fedja | ---Pointwise may be nice mathematical myths, locally uniform is closer to what we do in practice, especially on an elementary level--- OK, now it is "locally uniform", not just "uniform". That is a great step towards where we can possibly converge. How exactly do you suggest to define "local uniformity" to the students? | |
| Feb 6, 2011 at 16:08 | comment | added | fedja | ---I see mathematics as a messy bag of tricks to solve all kinds of messy problems, not as "a set of definitions and statements.--- Here we totally disagree. I see mathematics as a well-organized toolbox to solve all kinds of messy problems. "Definitions and statements" are just labels on various compartments and if you screw your labeling, you'll have hard time finding the right tool. | |
| Feb 6, 2011 at 16:04 | comment | added | fedja | ---Sorry, man, division is a messy operation, things get messy near singularities, nothing we can do about it--- OK, you finally admitted this with your approach. With the standard one, division is neat and clean. ---but I confront it earnestly by introducing the notion of locally nice, and you sweep it under the rug--- "Locally nice" is the same as epsilon-delta. And I do not "sweep it under the rug", on the contrary, I emphasize that delta depends on the point as well as on epsilon and make it clear in every proof what exactly determines it so one can easily trace uniformity if needed. | |
| Feb 6, 2011 at 7:59 | comment | added | Misha | @fedja:--At last, I do not see why uniformity is something that is always there in reality. When driving, the vehicle reacts to steering differently when the road is icy...---Global no, but local yes. You go point by point and then use compactness as a crutch, I go interval by interval, and also sometimes have to use compactness as a crutch. But I don't need compactness right away, and you do, because your definitions of continuity and differentiability are so weak. Pointwise may be nice mathematical myths, locally uniform is closer to what we do in practice, especially on an elementary level. | |
| Feb 6, 2011 at 7:09 | comment | added | Misha | @fedja:--At last, I do not see why uniformity is something that is always there in reality. When driving, the vehicle reacts to steering differently when the road is icy...---Global no, but local yes. You go point by point and then use compactness as a crutch, I go interval by interval, and also sometimes have to use compactness as a crutch. But I don't need compactness right away, and you do, because your definitions of continuity and differentiability are too weak. | |
| Feb 6, 2011 at 5:18 | comment | added | Misha | @fedja:---it fails to lead to a consistent set of definitions and statements.--- You state it as a proven fact, yet you haven't caught me with any inconsistency yet. I see mathematics as a messy bag of tricks to solve all kinds of messy problems, not as "a set of definitions and statements." When a definition doesn't fit, you adjust it. Trying to fit too much into one definition makes it so broad that it doesn't mean much any more | |
| Feb 6, 2011 at 4:38 | comment | added | Misha | @fedja:---In general, I want the following. There is a class of nice functions that is closed under standard arithmetic operations (+,-,*,/) and composition with the usual agreement about the domain of the result of each operation.---Sorry, man, division is a messy operation, things get messy near singularities, nothing we can do about it, but I confront it earnestly by introducing the notion of locally nice, and you sweep it under the rug. | |
| Feb 6, 2011 at 4:29 | comment | added | Misha | @fedja:--If you resort to "$1/g$ is uniformly Lipschitz on every set where $g$ is uniformly bounded away from $0$ etc. when trying to say that $1/x$ is nice on $(0,1)$ and the whole thing becomes a mess --- No, I am no saying that $1/x$ is nice on the whole $(0,1)$, it becomes a mess near $0$, I will have to use locally uniform notions here. Things become messy near singularities, and even you will have to resort to some tricks, like improper integrals, to handle it. | |
| Feb 6, 2011 at 4:06 | comment | added | Misha | @fedja: ---IMHO, one hard definition is easier to fight through than it is to memorize 10 easy caveats.--- Memorize? I thought we wanted to cultivate understanding, not memorization and regurgitation of definitions and theorems. Easier to memorize, but harder to comprehend. It looks like you prefer mathematics as some sort of monotheistic religion where everything is neat and clean, but the world (and even mathematics) is messy, and you better get used to it. | |
| Feb 6, 2011 at 1:28 | comment | added | fedja | At last, I do not see why uniformity is something that is always there in reality. When driving, the vehicle reacts to steering differently when the road is icy but you still can guide it on virtually every road if you choose your speed right depending on the road condition and turn sharpness. Somehow the students get the idea that your delta always exists but depends on both x and epsilon on the road (well, perhaps, some don't, but they get eliminated naturally). Why cannot they learn this in a math. class? | |
| Feb 6, 2011 at 1:23 | comment | added | fedja | And, yes, I want a consistent presentation, not just the one that conveys the ideas but fails to put the details straight. You say that my consistency is illusory because it fails to embrace uniformity, but I can also say that your uniformity is illusory because it fails to lead to a consistent set of definitions and statements. | |
| Feb 6, 2011 at 1:14 | comment | added | fedja | The trade here is between learning a harder definition once and having nothing to worry afterwards and learning an easy definition and getting many caveats later (like $x^2$ is defined on the entire line but not "good" there (but $x$ is and you cannot even multiply without worry, as it turns out), the division does not result in a "good" function on the natural domain, roots cannot be taken in Lipschitz category and the inverse function is out of control no matter what you do). IMHO, one hard definition is easier to fight through than it is to memorize 10 easy caveats. | |
| Feb 6, 2011 at 1:07 | comment | added | fedja | In general, I want the following. There is a class of nice functions that is closed under standard arithmetic operations (+,-,*,/) and composition with the usual agreement about the domain of the result of each operation. "Continuous at each point of the domain" is such a class. "Uniformly Lipschitz/Holder" on the domain is not. | |
| Feb 6, 2011 at 1:00 | comment | added | fedja | ---When you pick your epsilon, your delta will go to zero near zeroes of $g$--- It seems like there is some impenetrable wall between us here. My division theorem is "if $g$ is continuous at each point, $1/g$ is continuous at each point where $g$ is not $0$" and that is OK because the definitions fit together. What is your statement here? If you resort to "$1/g$ is uniformly Lipschitz on every set where $g$ is uniformly bounded away from $0$", you'll have to introduce exhaustions, etc. when trying to say that $1/x$ is nice on $(0,1)$ and the whole thing becomes a mess. | |
| Feb 5, 2011 at 15:11 | comment | added | Misha | fedja, on Jan 26 at 4:41 you also said: "As to the IVT, I use bisection far more often than Newton when I need real roots of something." ---The bisection method works fine in the uniformly continuous theory, and the argument justifying it does not need IVT. The critique in Bridger's book says that you may have more and more troubles deciding the sign of $f(x)$ when $f(x)$ gets small, and end up never getting your answer to any good accuracy. In fact, if you disregard Bridger's objections (which is the classical way), IVT for continuous functions stay valid (with its proof). | |
| Feb 5, 2011 at 14:45 | comment | added | Misha | @fedja, on Jan 26 at 4:41 you said: "The existence of extrema theorem is the basis for Rolle, Rolle is the basis for mean value and second order Taylor and those are bread and butter for all estimates from determining concavity to the Newton method." In my response on Jan 26 at 5:50 I presented a weaker form of the Rolle, but there is a better answer. We don't need Rolle because we can derive the estimates based on it by using the monotonicity principle that says that a function with non-negative derivative is non-decreasing and that is proven directly in our (locally) uniform approach. | |
| Feb 4, 2011 at 18:10 | comment | added | Misha | Fedja, the key word in your statement is "consistently." To keep your estimates for $1/g$ uniform you need $g$ to be bounded away from zero, and it is the same way in classical theory. Yes, $1/g$ is continuous at every point where $g$ is continuous and not zero, but this generality is illusory. When you pick your epsilon, your delta will go to zero near zeroes of $g$. We have already agreed on all that. You say it is all obvious. Good, let's move on to something not obvious. | |
| Feb 4, 2011 at 13:30 | comment | added | fedja | ---We can do plenty with Lipschitz, Hoder etc. What are you still arguing about?--- I claim that we cannot consistently do division with the uniform Lipschitz/Holder and I'll stick to it unless you show how to do it. Your own notes avoid the discussion of division altogether. ---What were your difficulties? Maybe I can help you to deal with them--- Isn't this funny? I'm telling you exactly what they were one by one and you just say "it can be done in general" and show only the parts that are obvious. | |
| Feb 3, 2011 at 20:23 | comment | added | Misha | @fedja: Please move on to your next concern, I think we're pretty much done with composition and division, the estimates just deteriorate when you get close to the singularities. If you disagree, what would you want me to clarify? You said you once tried to develop calculus/analysis in a similar way, but gave up. What were your difficulties? Maybe I can help you to deal with them. | |
| Feb 3, 2011 at 17:48 | comment | added | Misha | I have been away for a few days because i was notified that my account had been suspended till 3/2/2011. Looks like it's back on now, at least I can add comments, although the "edit' option of my answer is not there. Meanwhile, Ursula has translated 3 (out of 10) sections of Karcher's lecture notes, I have edited them to be more readable. I'm still having troubles with pictures, as soon as I get them right, I will start putting them up somewhere. | |
| Feb 3, 2011 at 17:21 | comment | added | Misha | @fedja:I thought I have explained already how to proceed in my comment on Jan 28 at 6:56. I don't intend to write a detailed treatise in comments. As you said on Jan 27 at 2:15, "epsilons and deltas are allowed "to deteriorate" (i.e., to be not uniform)." I agree, and you have to take this deterioration into account when you do numerical calculations. What's the big difference? Sometimes we have to work with specific moduli of continuity, and sometimes less specific classical notions are fine and even preferable. We can do plenty with Lipschitz, Hoder etc. What are you still arguing about? | |
| Feb 1, 2011 at 16:33 | comment | added | fedja | Misha. If you claim that all things can be ironed out, iron out division and composition simultaneously so that everybody can see a clear presentation, not just patches that are more or less obvious. Then I'll move to the next concern. I believe that you'll have to introdice some version of epsilon-delta at this level already and this is the very beginning. I tried to make a consistent presentation in a similar spirit but failed, so I'm not buying your "details can be made fine" statement. Sorry for my disappearance for the weekend :). | |
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| Jan 29, 2011 at 8:33 | comment | added | Misha | @Steven:---how do you talk about the total derivative of a multivariable map--- The basic estimate works fine, you only have to replace the absolute value with some norm. That's what you could call "locally uniform linearity." You can also look at it as factoring, see the section on many variables from my paper referred to in my answer, it works out really nicely. A big part of calculus is the interplay between algebra and geometry. You often can use one to understand the other, but you need both of them. Don't take this learning style doctrine too rigidly. Good luck to you too. | |
| Jan 29, 2011 at 7:52 | comment | added | Misha | @Steven: I am glad you are doing the right thing explaining some linear algebra. As for Stewart, it has a lot of nice problems in it, but the theory is awful. What's more, it is mostly irrelevant, there is a bad disconnect there. I'd suggest exposing the ideas the way you like and your students can understand, and then using Stewart as a problem book. That's what mostly happens in practice. It's like that exercise in the first edition of Lang's Algebra: "Take any book on homological algebra and prove all the theorems without looking at the proofs in the book." | |
| Jan 29, 2011 at 7:26 | comment | added | Misha | @fedja:---It is all not about whether your suggestion has good sides or not...----Sorry, I missed that. I don't suggest to totally expunge limits and continuity from all the analysis. I suggest to start with a lighter and more elementary tools. Taking your analogy, you can use a biplane or a hang glider when it's possible, you also don't have to drive a MAC truck to a grocery around the corner, you can walk or ride a bike. Also limits are the tools of mathematical theorizing, and uniform estimates are much more useful in practical calculations. Drop this one-size-fits-all ideology and enjoy. | |
| Jan 28, 2011 at 20:05 | comment | added | Steven Gubkin | Might I also say that I think my approach is more geometric and your approach more algebraic. So I think the students learning style is going to have a big impact on which one is "much easier to understand". | |
| Jan 28, 2011 at 20:00 | comment | added | Steven Gubkin | btw - how do you talk about the total derivative of a multivariable map from your perspective? "Locally linear approximation" is really the only way I have ever thought about it. | |
| Jan 28, 2011 at 19:54 | comment | added | Steven Gubkin | I think that a lot of instructors lack the bravery to step outside of the norm and get creative in their presentation of concepts. Some departments actively discourage this. I am somewhat saddened that you interpreted my praise as a "justification of my own inaction". Best of luck to you. | |
| Jan 28, 2011 at 19:51 | comment | added | Steven Gubkin | that makes sense to you. Clearly Stewart did not think that having linear transformations around was neccisary, and I do not doubt that he does a very good job teaching the material out of his own book. But I cannot teach the material well the way that he presents it - I have to present it from my own perspective. I am trying to suggest that I would not be able to teach differentiation the way you do, because it does not speak to me or inspire me. But, as it clearly does inspire you, there is a good chance that you could teach students a lot this way. | |
| Jan 28, 2011 at 19:48 | comment | added | Steven Gubkin | I am not trying to justify my inaction. Quite the contrary. In my own teaching, I am very careful to present things the way I actually understand them. For example, I am currently teaching a multivariable calculus class. The textbook (Stewart) does not present any information on linear transformations, but I believe very strongly that you cannot understand the derivative without first understanding linear maps. So I have written some handouts presenting this point of view to my students. In fact, I was trying to say that I admire your bravery for stepping outside the norm to teach in a | |
| Jan 28, 2011 at 19:22 | comment | added | Misha | @Steven Gubkin.---So I think the presentation of material is really kind of irrelevant - especially if students are not going on in mathematics.--- You are absolutely wrong, the simpler the material -- the more likely the students to follow. Factoring $p(x)-p(a)$ into $x-a$ is much easier to understand than limits. You are just trying to justify your inaction. | |
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| Jan 28, 2011 at 17:10 | comment | added | Steven Gubkin | teach the subject your way, or that it would help for people to try. The goal of a lecturer is to guide students down a path to understanding, and as a lecturer you cannot use any understanding other than your own. So I think the most important thing is to let lecturers present the material in their own way, so that students see natural thought. | |
| Jan 28, 2011 at 17:08 | comment | added | Steven Gubkin | I just don't think that limits are that much of a barrier to most students. I personally think the greatest barrier is a system which encourages memorization over thought. No matter what content you teach (as long as what you say is true) getting students to think is the most essential thing. So I think the presentation of material is really kind of irrelevant - especially if students are not going on in mathematics. The goal should just be to get students to think mathematically. If you can do that with your approach - more power to you. But I don't think that everyone would be able to | |
| Jan 28, 2011 at 16:16 | comment | added | Misha | @Steven I gave a 2 hour lecture to high school students in China in 2009, pretty much following the talk at mathfoolery.com/talk-2004.pdf and they said it was too easy. It is true that many students just memorize the formulas. These students will have a better chance if you start by demonstrating these formulas for polynomials and then saying that the rules work in general, using them for problems of interest and then explaining some Lipschitz stuff, using polynomials as a motivation. Doesn't it look more reasonable than hitting them on the head with limits? | |
| Jan 28, 2011 at 16:01 | comment | added | Misha | @Steven Unfortunately not at the college level. I have done some volunteer teaching of interested high school students at MIT, they seemed to be rather pleased with my approach to the subject. Trying to push them into figuring things out for themselves didn't work that well, though, although a few liked it. Mark Bridger and Hermann Karcher (see the main body of my answer) taught it to engineering students and reported good results. I know a tutor that used my approach to explain calculus to a business student with no brain for math, and he got A, so some evidence is there. | |
| Jan 28, 2011 at 15:30 | comment | added | Steven Gubkin | @Misha Have you actually tried teaching this way? Do you really think that intro calc students who have "no desires to become mathematicians" will absorb any of this? I ask because I think it often happens that students completely ignore the theoretical underpinnings of the course and simply memorize formulas. If your students are doing this, I see no reason to present it your way over the standard way. What would be exciting is if you find that your approach somehow forces a greater percentage of the class to actually think about the mathematics. | |
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| Jan 28, 2011 at 7:31 | comment | added | Misha | All these small details you are concerned with can be ironed out. It can be done by the students in the problem sets that would be rather elementary, still would teach them how mathematics is done. The point of teaching mathematics is give the students an opportunity to do it, not to give them all the details. If you like the ideas, you can make them work. I am a little annoyed that you would not reveal your identity. | |
| Jan 28, 2011 at 7:14 | comment | added | Misha | Again, I am mostly concerned about introductory calculus for people who not necessarily plan to become mathematicians. Polynomials and Lipschitz give you a good start, Holder helps if you insist on going over algebraic singularities, more refined generalities can be treated later in the course (when the students get more eexperience) or relegated to more advanced courses. Read Karcher's English summary. I am also concerned about high school students who are not ready for the abstractions, but would be fine with a more elementary,but still mathematical treatment. | |
| Jan 28, 2011 at 6:56 | comment | added | Misha | Composition: Lip(Lip)=Lip, Hol(Lip) and Lip(Hol) is Hol with the same power, Hol(Hol) is Hol with a different power, UC(UC)=UC with a different modulus of continuity. Near singularities, your constants deteriorate. Division: constants deteriorate near zeroes of the divider, away from them you are fine. It is the same way with classical theory if you keep track of your epsilons and deltas. I (and Karcher) suggest doing general continuity later in the course, when Lip and maybe Hol (and maybe power series) are well understood. On compactness: students know about close intervals, it's enough. | |
| Jan 28, 2011 at 6:25 | comment | added | fedja | @Mariano Yeah, if I could only figure out what this particular phrase was supposed to mean... Nevermind, I'm not "touchy" :) @Misha It is all not about whether your suggestion has good sides or not. Nobody objects that something in this spirit would be nice (like it would be nice to float in the air over an emerald forest at dawn, supported above the ground just by your desires and not by some clumsy, noisy, and stinky machinery). The issue at stake is the technical possibility. The "classical" solution is to learn to live with and to operate the machinery and to enjoy the flight then. | |
| Jan 28, 2011 at 6:03 | comment | added | fedja | You seem to misunderstand me. At this moment, the students know neither power series, nor compact sets, nor convergence. All they know is what you have told in your introduction. The classical way is to tell the definition of continuity at a point (one basic notion) and then to demonstrate that it survives arithmetic and composition (so "normal" operations on nice functions result in nice functions). So far the classical lecture is smooth and consistent. You started with several classes out of which only 2 can be defined right away. Now you face division and composition. How to present those? | |
| Jan 27, 2011 at 23:57 | comment | added | Mariano Suárez-Álvarez | «I probably understood him a bit better than he cared for...» Well, that's surely going to attract good will! | |
| Jan 27, 2011 at 23:33 | comment | added | Misha | Emerton wrote me in an e-mail: In my own education, I know that notions like Lipschitz and other concepts that focus on uniformity and rates of convergence came late, and I didn't appreciate them for a long time; on the other hand, in applications it seems that having an awareness of the role of uniform bounds, or quantitative estimates in general, is often very important, and what I like about your approach is that it brings these concepts out right away. | |
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| Jan 27, 2011 at 22:05 | comment | added | Misha | fedja:---You certainly turned some people away (Pete, say)--- Oh, well, I probably understood him a bit better than he cared for... I actually e-mailed him my apologies with some references and the files he could not access. He hasn't replied yet, some people are touchy.... | |
| Jan 27, 2011 at 21:55 | comment | added | Misha | @fedja:---you give up on analytic and locally uniformly continuous.--- No, converging power series are locally Lipschitz on their convergence intervals. "Locally" means that "on any closed subinterval," (or, more generally, on any compact subset). This can be done by algebra of power series and checking the convergence. Since I have explained what "locally" means, I don't give up on locally uniformly continuous either (well, "uniformly" is a bit redundant here, right?).---about division?--- $f/g$ is locally nice on the set where $g \ne 0$. "Locally" allows for deterioration of the constants. | |
| Jan 27, 2011 at 19:21 | comment | added | fedja | As to your second question, I believe that yes, this is being read, and, moreover, you, probably, have about as much attention now from wide mathematical audience to your teaching ideas as you are ever going to get. You certainly turned some people away (Pete, say) by the moment and what happens next depends on what you say, but I find public discussions more useful than private e-mails (full public record of speeches and independent arbitrage are big pluses. Besides, you can really win some people you don't know to your side). Of course, if you do not care, we can stop here. | |
| Jan 27, 2011 at 19:10 | comment | added | fedja | Wait a bit... So, you give up on analytic and locally uniformly continuous. You are left with Polynomials, uniformly Lipschitz on the domain and uniformly continuous with modulus omega. Let's say that on this first run you just say that we can replace $|x-y|$ with some other similarly behaving $\omega(x-y)$ to be specified later. You see, I'm just trying to put one lecture together and see if your method can survive it. The next thing is arithmetic. The classical base survives all 4 arithmetic operations with natural prohibition to divide by 0. What will you say about division? | |
| Jan 27, 2011 at 16:51 | comment | added | Misha | Do you think anybody besides you and me is reading this? If not, it's sort of a waste, do you want to e-mail me, and then we can decide where to continue? | |
| Jan 27, 2011 at 16:38 | comment | added | Misha | @fedja:---What definition(s) will you use?---Polynomials are just that, and they are Lipschitz, $|f(x)-f(a)|\le L|x-a|$, uniformly in $x$ and $a$. For any other modulus of continuity $m$ we take $|f(x)-f(a)|\le Lm(|x-a|)$. If take the union of all these classes, we get the class of all the uniformly continuous functions. The fact that pointwise continuous functions on compact sets are uniformly continuous is a matter of compactness. I anticipate that you will say that the constant in the inequality will deteriorate near singularities, but we have already discussed that, right? | |
| Jan 27, 2011 at 16:20 | comment | added | Misha | @fedja:----The classics is "continuous at a given point".---You got it! Now, for polynomials, $x-a$ divides $f(a)-f(a)$, and the ratio, evaluated at $x=a$, gives you $f'(a)$. If he same is true in the class of functions continuous at $a$, you call $f$ differentiable at $a$. Differentiation is just factoring! Mark Bridger in his 2007 book defines $f$ as uniformly differentiable if this factoring holds in the class of functions uniformly continuous in both $x$ and $a$. We can play the same game with Lipschiz, Holder, or any other class of functions admitting some given modulus of continuity. | |
| Jan 27, 2011 at 16:18 | comment | added | Misha | So, fedja, where do you want to take our discussion, if you want to continue? | |
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| Jan 27, 2011 at 13:43 | comment | added | Misha | @fedja yesterday---By the way, once you have continuity, you can say that $f'(a)=L$ if the function defined as the difference ratio for $x \ne a$ and $L$ for $x=a$ is continuous at $a$, if you want.–--Yes, absolutely, and even more, $\lim_{x \rightarrow a} f(x)=L$ if the function defined as $f(x)$ for $x \ne a$ and $L$ for $x=a$ is continuous at $a$. Continuity first, like E.Cech did it in the 30s (according to Jerry Uhl)! | |
| Jan 27, 2011 at 11:53 | comment | added | fedja | OK, we'll discuss the philosophy later (don't worry, I'll not try to escape from it; I just do not want to create a thread with two things in parallel: nobody will be able to read it). ---So, what base class of nice functions will you start with---Polynomials, Lipschitz, analytic, Holder, (locally) uniformly continuous.--- Fine. The next thing is to formally define your class so that the students can recognize the membership. The classics is epsilon-delta. What definition(s) will you use? (you are always welcome to reduce your list any time you want but keep it non-empty). | |
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| Jan 27, 2011 at 5:12 | comment | added | Misha | @feddja: Here are a some references for untraditional treatments of calculus and elementary analysis: 1)Real Analysis: a Constructive Approach by Mark Bridger, 2007 2)Ch.1&2 of Analysis by Its History by Hairer and Wanner, 1996. 3)Lecture notes by Hermann Karcher (in German with an English summary, hopefully will be translated into English soon) at math.unibonn.de/people/karcher/MatheI_WS/ShellSkript.pdf | |
| Jan 27, 2011 at 4:58 | comment | added | Misha | @fedja:---real analysis training you proposed---I am more concerned about calculus on an elementary level.---Weak students?...the task is to bring them up to the level of modern science, not to reduce the modern science to their level---You call pointwise notions "modern science?" To me they look like a dubious myth from 19th century. Maybe we can use the ideas of modern mathematics to make calculus simpler? That's what I have been trying to do.---So, what base class of nice functions will you start with---Polynomials, Lipschitz, analytic, Holder, (locally) uniformly continuous. | |
| Jan 27, 2011 at 4:24 | comment | added | Misha | @fedja:---The epsilons and deltas are allowed "to deteriorate"---And you have to take it into account when you calculate with such functions. The classical theory sweeps it under the rug. ---Give me an example---Different moduli of continuity prepare strong students to come to grips with variety of functional spaces that are useful in PDEs, for example. It shows them that definitions are not carved in stone but should be flexible, depending on the problem. Differentiation as factoring draws parallels with algebraic geometry and functional analysis.---Practical---Stock market fluctuations | |
| Jan 27, 2011 at 3:27 | comment | added | Emerton | I found this post very stimulating, and voted it up. | |
| Jan 27, 2011 at 2:32 | comment | added | fedja | Weak students? Yes, but the task is to bring them up to the level of modern science, not to reduce the modern science to their level. "Grandiose mathematical theorizing" is there for a reason and the reason is that, for all we know, the nature loves complex structures more than simple ones. But let's go back to your approach and restrict ourselves to the basic real analysis. You want to explain to your students that some functions are nice and some aren't. So, what base class of nice functions will you start with? The classics is "continuous at a given point". | |
| Jan 27, 2011 at 2:15 | comment | added | fedja | The epsilons and deltas are allowed "to deteriorate" (i.e., to be not uniform) by the nature of the double quantifier definition and that is the whole point of having them the way they are. Anything practical, you say? Let's discuss it. Give me an example of a practical problem for which the sort of real analysis training you proposed would be more beneficial than the traditional one. Or just any example of a practical problem you dealt with. Strong students do turn off for many reasons but epsilon-delta definitions is certainly not one of them. They can digest it. | |
| Jan 26, 2011 at 18:17 | comment | added | Misha | @fedja: Now, how about $1/(x+x^5−1)$? Stay away from singularities if you want to stay uniform, your epsilons and deltas also deteriorate there. One-size-fits-all definitions is a mathematical fallacy, it may be good for grandiose mathematical theorizing, but when you get to anything practical or more specialized, they don't fit that well. | |
| Jan 26, 2011 at 18:07 | comment | added | Misha | @fedja: ----It kills the subject to start with them.---- Not the "subject" but the "weak students". 1)Don't you want the weaker students to understand it too, maybe not all the technicalities, but the essence of it, so they could use it in their further studies/research? Then why push the inessential technicalities first? 2)----The subject is still in fairly good health :) ----Not really, those weak students are smarter than you think, and many strong students get turned off and decide not to pursue mathematics. A prof. from Kyoto complained that smart kids don't go into math any more | |
| Jan 26, 2011 at 12:10 | comment | added | fedja | Also, Misha, you don't understand Pete at all. When teaching, we have to take care not of the "big picture" (I have a pretty clear one of the entire undergraduate analysis sequence) but of "pesky details". The standard approach is tuned up in this respect. The one you propose is not (or, at least, not yet). Pete just wanted to see if some points that are clearly out of tune can be tuned without putting the already tuned ones out of tune. I want to see pretty much the same. | |
| Jan 26, 2011 at 12:03 | comment | added | fedja | ----It kills the subject to start with them.---- Not the "subject" but the "weak students". The subject is still in fairly good health :). Now, how about $\sqrt[3]{x+x^5-1}$? We'll need to use intervals not coming close to the (unknown!) zero to have uniform estimates. Suppose I just want to explain that it is differentiable wherever it is not $0$. That is not a fine point but a very basic claim. How does it reconcile with your definitions? I cannot say that it is "uniformly differentiable" there (it is false) and I cannot explicitly tell the admissible domains of uniformity either. | |
| Jan 26, 2011 at 11:34 | comment | added | Pete L. Clark | @Misha: please don't say that I don't want to understand the ideas. I have said multiple times that I do want to do so, so suggesting otherwise is essentially accusing me of intellectual dishonesty and/or bad faith. That's pretty insulting given the demonstrable amount of time I have put into asking and answering questions on this website. If you want to claim I am not competent to follow your arguments: that's fine; go ahead. Perhaps you could clarify what your audience is, though, if you do not intend your calculus writings to be easily understandable by a PhD mathematician. | |
| Jan 26, 2011 at 9:08 | comment | added | Misha | Sorry, you don't come across as a person who wants to understand the ideas and is willing/able to fill in the gaps from hints and context. You come across as a person who picks on formal minutia. | |
| Jan 26, 2011 at 8:29 | history | edited | Misha | CC BY-SA 2.5 |
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| Jan 26, 2011 at 8:02 | comment | added | Pete L. Clark | @Misha: If you interpret "What do you mean when you write X?" as picking on you, we cannot have a conversation. It's also distressing that you don't find the fact that a professor of mathematics finds your writings on calculus hard to follow as being anything other than criticism. As I said, I came to this answer with some interest in your point of view. After receiving your reaction, this is no longer the case. | |
| Jan 26, 2011 at 7:31 | comment | added | Misha | But the existence of such $z$ in MVT is very mathematically subtle, although intuitively obvious. In the Lipschitz theory everything is explicit, you get naturally from polynomials to it, it is still earnest mathematics, not hand-waving. | |
| Jan 26, 2011 at 7:24 | comment | added | Misha | Look, most students want to learn how to use calculus in solving problems in the fields of their interest, they care little about fine points of mathematical ideology. After they get some experience on the elementary level and develop some intuition and skills, they can refine their ideology when needed. All the problems of substance together with the solutions stay the same, it's the ideology that is rearranged a little. Uniform estimates correspond to our intuition and work much better than pointwise derivatives and pointwise continuity by limits. It kills the subject to start with them. | |
| Jan 26, 2011 at 7:17 | comment | added | fedja | ----Compare the explanation of $f$ increasing if $f'>0$ in the standard and simplified approach and see the difference!---- OK, let's do it. Standard: $f(y)-f(x)=f'(z)(y-x)$ (MVT) and the product of two positive numbers is positive. What's "simplified"? ----If you are still in the Boston area we can meet.---- Sure, the only little problem is that I am about 900 miles away from Boston. | |
| Jan 26, 2011 at 7:11 | comment | added | fedja | ----If you teach them after you work with, say Lipschitz estimates, the students will have a much better chance.---- Sure. Have I ever claimed the opposite? I only claimed that one needs to do it, not that it should be done during the first five minutes of the first lecture. Two-three Lipschitz theorems that do work well can easily be done first, as I said, but I prefer to pass from that to standard continuity. By the way, once you have continuity, you can say that $f'(a)=L$ if the function defined as the difference ratio for $x\ne a$ and $L$ for $x=a$ is continuous at $a$, if you want. | |
| Jan 26, 2011 at 7:11 | comment | added | Misha | You can solve $x^{2/3}=y$ explicitly, $x=y^{3/2}$ most students will have no problem with it, especially after they are done with Lipschitz-based theory. | |
| Jan 26, 2011 at 7:05 | comment | added | Misha | You don't need the second part to work with piecewise analytic functions. If you are still in the Boston area we can meet. | |
| Jan 26, 2011 at 6:52 | comment | added | Misha | About 90% of the students will never use limits as such, and they will never understand them. If you teach them at the beginning. If you teach them after you work with, say Lipschitz estimates, the students will have a much better chance. Don't forget about convergence of the series etc. These are the limits you can not avoid. I am just against teaching pointwise derivatives as limits and doing generalities before examples. Compare the explanation of $f$ increasing if $f'>0$ in the standard and simplified approach and see the difference! | |
| Jan 26, 2011 at 6:46 | comment | added | fedja | ---You don't have to define them before you use the concrete examples, such as Lipschitz and Holder. --- Hah? It is not about "before" or "after". It is about whether you can "ever" do it consistently. I'm totally fine with doing a few exercises with Lipschitz functions first (sum, product, composition, etc.), then asking what exactly was important about that $|x-y|$ that popped up everywhere, but at this stage epsilon-delta come into play and I redo all we did with the standard definition. This trick does work and I do not argue about that. The question is how to avoid the second part. | |
| Jan 26, 2011 at 6:38 | comment | added | fedja | ----defining first that an increasing function is continuous if it doesn't skip any values---- OK, I'll sort of buy that, but by now it is a full fledged analysis course, just turned inside out: what is usually taught as various consequencies of continuity becomes a sophisticated definition of it. And how exactly do you propose to check that $x^{2/3}$ doesn't skip values? | |
| Jan 26, 2011 at 6:37 | comment | added | Misha | You don't have to define them before you use the concrete examples, such as Lipschitz and Holder. Subadditivity of the moduli of continuity eliminates the nightmares, I have worked it out. | |
| Jan 26, 2011 at 6:33 | comment | added | Misha | Well, you are saying essentially that we can not deal with any approximations without dealing with limits, and I am not buying it. Moreover, jumping into limits, or paying a lip service to them by using the notation before students learn anything of substance, is much worse than introducing them when they become necessary and after the simpler instances of approximation are well digested. | |
| Jan 26, 2011 at 6:33 | comment | added | fedja | Tracking arbitrary moduli in compositions is a nightmare, besides, as I said, you cannot even define the abstract modulus of continuity, so we've got to stick to Lipschitz, which doesn't allow even to take square roots of non-negative functions. You can, probably, figure out how to teach each particular thing (except the inverse function) this way, but I'm very far from being convinced that it'll be consistent as a whole and will not be a dead end if someone decides to extend his knowledge of analysis. It still seems cheaper to me to spend 2 lectures on the standard definition of the limit. | |
| Jan 26, 2011 at 6:24 | comment | added | Misha | @fedja: There is a nice way to treat continuity by (Susan Bassein, An Infinite Series Approach to Calculus) defining first that an increasing function is continuous if it doesn't skip any values (i.e. you can draw its graph in 1 stroke). Then you can use it as a modulus of continuity to define continuity in general. Of course you want your moduli of continuity to be nice (say, subadditive) for the theorems to be pretty. But in practice (especially in introductory calculus) Holder estimates take cake of almost anything. | |
| Jan 26, 2011 at 6:21 | comment | added | fedja | "As small as you want" is exactly the expression you tried to eliminate. Once you start saying "there exists $a$ such that $f'(a)$ is as small as you want", I see no difference with "for every $\varepsilon>0$..." and you defeat the whole purpose of the alternative exposition. Moreover, in place of one standard "for every ..., there exists...", you introduce several nonstandard ones. The main aim of the whole exercise is to kill this construction, not to multiply it. | |
| Jan 26, 2011 at 6:11 | comment | added | Misha | @fedja: Locally uniform differentiability with a given modulus of continuity makes it perfectly clear what local approximate linearity means. It is the basic estimate from our definition of the derivative! Where there is (uniform) epsilon-delta, there is a (uniform) modulus of continuity, differentiability, etc. Of course, tracking the moduli may be cumbersome, but at the beginning they work well. Hermann Karcher of Bonn University taught an intro analysis like this, but he jumped from Lipschitz to general uniform continuity. Ask Dick Palais. I am in Cambrige, MA if u r @BU we can meet. | |
| Jan 26, 2011 at 5:50 | comment | added | Misha | @fedja: About IVT: a very similar result holds for a Lipschitz (or any given modulus of continuity) function. It says that by bisection you can always get an approximate solution $a$ to $f(x)=0$, not in the sense that if is close to the exact solution, but in the sense that $f(a)$ is as small as you want. The approximate Rolle also holds in a similar sense, that you can find $a$ such that $f'(a)$ is as small as you want. It takes care of both of your objections. You can see a nice critique of IVT in the 2007 book "Real Analysis: a Constructive Approach" by Mark Bridger (pages 67 and 163). | |
| Jan 26, 2011 at 5:13 | comment | added | fedja | Another thing forget is that in the differential calculus the task is not to pass down the differentiation tool but to convey the general idea of "local approximate linearity", which you can't do without explaining what the words "local" and "approximate" mean, which inevitably leads to some version of the epsilon-delta language. I gave up on an idea similar to yours (it is a natural reaction to BC students and you are by no means the first who came up with this) not as much because I had problems with some particular issue, but because I couldn't put the whole thing together. | |
| Jan 26, 2011 at 4:58 | comment | added | Misha | @Pete L. Clark: It was a parenthetical remark of a philosophical nature that is not used anywhere else in the post. I didn't mean to supply all the details. Picking on it is totally unfair. The hint I gave you, that we approximate the derivatives, should be sufficient for anybody who took an introductory analysis to figure it all out. I assumed that it would be enough for you, since you are a professor of mathematics. | |
| Jan 26, 2011 at 4:41 | comment | added | fedja | The little problem is that those "pet existence theorems" I mentioned are actually very "computational". The existence of extrema theorem is the basis for Rolle, Rolle is the basis for mean value and second order Taylor and those are bread and butter for all estimates from determining concavity to the Newton method. If you don't have Rolle, you are stuck with "there exists some constant", which is computationally useless. As to the IVT, I use bisection far more often than Newton when I need real roots of something. | |
| Jan 26, 2011 at 4:31 | history | edited | Misha | CC BY-SA 2.5 |
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| Jan 26, 2011 at 3:14 | comment | added | Pete L. Clark | ..."Then everything works out." Again, because this is a very vague statement, I don't understand at all what you mean. Don't you want me to understand what you mean? If not, why are you posting here? | |
| Jan 26, 2011 at 3:13 | comment | added | Pete L. Clark | @Misha: Your response is very disappointing. I am trying to understand what you're writing. I would have thought that as an author that was your goal: to be understood. When I say that more precise definitions would improve my understanding, I mean just that. Regarding Weierstrass approximation: once again, derivatives do not appear in the statement of this theorem, and the metric involved is not one which guarantees anything about convergence of the derivatives. If you mean a result involving uniform convergence of the derivatives, please say so... | |
| Jan 26, 2011 at 3:09 | comment | added | Misha | @Pete L. Clark The estimate is uniform (in some interval), and a uniform estimate can be taken for a definition. Look, if you try to understand the ideas, look for the ideas, not for some petty inaccuracies that can be corrected in a rather obvious way. From what I do with this inequality it's obvious that it meant to be a uniform estimate in both x and h. I write for people, not for computers. On Weierstrass: we approximate uniformly the derivatives, then everything works out, see any book where term-by-term differentiation of a sequence of functions is treated | |
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| Jan 26, 2011 at 1:49 | comment | added | Pete L. Clark | @Misha: it does? Could you give a precise statement and a reference? I take it you are aware of the fact that uniform convergence of a sequence of functions does not imply convergence of the sequence of derivatives. | |
| Jan 26, 2011 at 1:47 | comment | added | Pete L. Clark | @Misha: I would like to understand some of your ideas, but your writing (both in this answer and in the linked documents I was able to access) is rather obscure to me. Is there some reason you do not write things more carefully? For instance: "we promote our basic estimate to the definition status and call the functions that satisfy this definition (uniformly) Lipschitz differentiable (LD)." An inequality is not a definition: you need, for instance, quantifiers. It would be very helpful to include a sentence beginning "A uniformly Lipschitz differentiable functions is..." | |
| Jan 26, 2011 at 1:42 | comment | added | Misha | @Pete L. Clark: the Weierstrass approximation theorem tell us about differentiation of uniformly (=continuously) differentiable functions that differentiation rules still hold for them because they hold for polynomials. Sorry for broken link, will fix it asap. | |
| Jan 26, 2011 at 1:25 | history | edited | Misha | CC BY-SA 2.5 |
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| Jan 26, 2011 at 1:16 | comment | added | Pete L. Clark | Also, the link in your third paragraph does not work for me. | |
| Jan 26, 2011 at 1:12 | comment | added | Pete L. Clark | What does the Weierstrass approximation theorem tell us about differentiation of continuous functions? | |
| Jan 25, 2011 at 23:15 | history | edited | Misha | CC BY-SA 2.5 |
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| Jan 25, 2011 at 21:06 | comment | added | Misha | Dear fedja, thanks for reading my article. Your impulse to dismiss it is understandable since the article makes your pet general notions of limits and continuity and your beloved pure existence theorems about intermediate values and attainment of maximum value look less relevant. But these belong to introductory analysis rather than calculus, and they are indeed rather irrelevant to the computational content of the subject. I will get to the inverse function theorem later, when I counter your objections one by one in the remarks I will add to my answer. | |
| Jan 25, 2011 at 12:44 | comment | added | fedja | As you noticed yourself, Lipschitz is not enough. You need all moduli of continuity. Now, the modulus of continuity (even in your exposition) is a function that is blah-blah-blah and continuous at 0. But what does that last condition mean if there are no limits or epsilon-deltas? I thought of teaching this way but the intermediate value theorem and the fact that every continuous function attains its maximum do not get any easier for Lipschitz functions and the inverse function theorem just fails miserably. | |
| Jan 25, 2011 at 0:30 | history | undeleted | Anton Geraschenko | ||
| Jan 25, 2011 at 0:30 | history | unlocked | Anton Geraschenko | ||
| Jan 24, 2011 at 15:18 | history | locked | S. Carnahan♦ | ||
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| Jan 23, 2011 at 15:32 | history | answered | Misha | CC BY-SA 2.5 |