Timeline for Why is differentiating mechanics and integration art?
Current License: CC BY-SA 4.0
52 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Sep 26, 2018 at 12:58 | history | suggested | Leo Oliveira | CC BY-SA 4.0 |
Updated dead link to Wikipedia's Leibniz integral rule page
|
| Sep 26, 2018 at 12:33 | review | Suggested edits | |||
| S Sep 26, 2018 at 12:58 | |||||
| Jun 11, 2014 at 11:05 | answer | added | user39115 | timeline score: 1 | |
| Oct 31, 2013 at 18:55 | review | Close votes | |||
| Oct 31, 2013 at 21:00 | |||||
| Oct 31, 2013 at 6:00 | answer | added | O.R. | timeline score: 7 | |
| Aug 21, 2013 at 12:44 | history | edited | François G. Dorais |
edited tags
|
|
| Aug 21, 2013 at 12:11 | history | protected | François G. Dorais | ||
| Aug 21, 2013 at 11:26 | answer | added | Juan I. Perotti | timeline score: 2 | |
| Oct 2, 2012 at 13:26 | answer | added | Peter Michor | timeline score: 15 | |
| Jun 19, 2011 at 20:28 | vote | accept | vonjd | ||
| Jun 7, 2011 at 18:09 | answer | added | Todd Rowland | timeline score: 1 | |
| May 31, 2011 at 1:16 | answer | added | Terry Tao | timeline score: 23 | |
| May 30, 2011 at 23:31 | comment | added | Deane Yang | Ryan (Reich), thanks for the reply. You're completely right about this, and I rescind my doubts. | |
| May 30, 2011 at 23:30 | answer | added | Deane Yang | timeline score: 10 | |
| May 30, 2011 at 22:14 | answer | added | gowers | timeline score: 116 | |
| May 30, 2011 at 21:33 | comment | added | Peter Luthy | @Deane A bit tongue in cheek, but: division of real numbers is harder than multiplication because it's secretly still multiplication, just with the additional step of inverting. Multiplication by $x$ is a continuous function, but multiplication by $x^{-1}$ is no longer continuous near zero. But if you look at, say, the circle group, multiplication and division are clearly equally difficult! | |
| May 30, 2011 at 20:20 | answer | added | Todd Trimble | timeline score: 53 | |
| May 30, 2011 at 20:18 | comment | added | Ryan Reich | @Deane: Power series; works for any function you would otherwise have trouble integrating symbolically. See my comment to Thierry Zell's answer. For fractions: prime factorization, like I said. But no matter how you write things, you give up some convenience (ease of composition/identification of closed form, or ease of addition) | |
| May 30, 2011 at 18:26 | comment | added | Deane Yang | I just got around to taking a closer look at Todd Trimble's answer. That (like Tao's) is a really good answer but again feels a little incomplete to me. In the end both his and Tao's answers seem to all come down to the fact that there are algorithmic ways to differentiate a function built out of old ones in standard ways (composition, product, quotient) but there are no analogous algorithms for integrals. This explains why differentiation is "easy", but I'm still not sure whether that explains why integration is "hard". I do like Todd's comparison to squaring and square-rooting. | |
| May 30, 2011 at 18:15 | comment | added | vonjd | @Deane: Thank you, I feel the same way: "[...] I think it is a lot more subtle and difficult than people are accounting for. Terry Tao's answer below is pretty good but still feels incomplete to me." - couldn't have said it better. | |
| May 30, 2011 at 18:06 | comment | added | Deane Yang | Ryan (Reich), I'm not sure I buy your argument. Do you have a different way to express functions/numbers that changes the situation you describe? | |
| May 30, 2011 at 18:04 | comment | added | Deane Yang | I concede that my comments were a bit too testy. Although I think the question is a valid and good one, I think it is a lot more subtle and difficult than people are accounting for. Terry Tao's answer below is pretty good but still feels incomplete to me. | |
| May 30, 2011 at 16:10 | comment | added | Ryan Reich | @Deane: For the same reason that integration is harder than differentiation: because of how we express our functions/numbers. Elementary functions are perfectly suited for differentiation because we have a rule for each operation. Likewise, writing numbers in decimal (or any) base is well-suited for multiplication because we have distributivity in both factors. If you want to divide easily, write a number as its prime factorization (though then you can't add). | |
| May 30, 2011 at 14:48 | answer | added | Terry Tao | timeline score: 85 | |
| May 30, 2011 at 11:38 | comment | added | Todd Trimble | Not meaning to pile on Deane here, but the question is more, "why is integration art and differentiation science?", rather than, "why is integration harder than differentiation?" The OP's second paragraph says more on this, and I came away thinking that the OP is more interested in the question of whether antidifferentiation algorithms exist or if not, why not. | |
| May 30, 2011 at 9:40 | answer | added | Denis Serre | timeline score: 15 | |
| May 30, 2011 at 6:52 | history | edited | vonjd | CC BY-SA 3.0 |
added 1 characters in body
|
| May 30, 2011 at 6:45 | history | edited | vonjd | CC BY-SA 3.0 |
clarification; added 7 characters in body
|
| May 30, 2011 at 6:05 | comment | added | Ryan Budney | @Deane, if that's a question you're interested in and think is appropriate for MO I encourage you to start a new thread. As-is, you seem to be derailing the discussion here. | |
| May 30, 2011 at 5:36 | answer | added | Peter Luthy | timeline score: 20 | |
| May 30, 2011 at 4:47 | comment | added | Deane Yang | Ryan, why is division harder than multiplication? | |
| May 30, 2011 at 4:06 | comment | added | Ryan Reich | @Deane: Which one is the inverse? It's not surprising to me that one should be harder, but you can still ask why integration in particular. | |
| May 30, 2011 at 3:57 | comment | added | Deane Yang | It seems to me that before we discuss differentiation and integraiton, could someone first explain why division is harder than multiplication? (Why should it be surprising that a process is easier to carry out than its inverse?) | |
| May 30, 2011 at 2:02 | answer | added | Thierry Zell | timeline score: 5 | |
| May 29, 2011 at 23:48 | comment | added | Dan Piponi | Although it's usually said that integration is harder than differentiation there are many senses in which the opposite true. As linear operators in functional analysis, integration is often much better behaved than differentiation. When doing exact real arithmetic, integration over an interval is computable but differentiation is not. (Eg. see homepages.inf.ed.ac.uk/als/Research/lazy.ps.gz) | |
| May 29, 2011 at 21:32 | comment | added | Ryan Budney | I think you really ought to qualify what you mean by "doable", since Riemann integrals exist in most "standard" axiom systems for mathematics. IMO the primary reason for the apparent "deep" nature of your question is largely due to improper formulation. If you talk about "doable" in terms of some class of elementary functions, then you see what the problem is, as has been mentioned by several people. | |
| May 29, 2011 at 20:58 | answer | added | Ryan Reich | timeline score: 5 | |
| May 29, 2011 at 20:39 | answer | added | Michael Renardy | timeline score: 8 | |
| May 29, 2011 at 20:28 | comment | added | Sridhar Ramesh | (Note that, for example, the product rule (mentioned in the original question) is just the instance of the chain rule as applied to the particular multivariable function of multiplication) | |
| May 29, 2011 at 20:17 | comment | added | Sridhar Ramesh | I think it just amounts to the existence of the chain rule (including in its multivariable forms) for differentiation, which makes differentiation easy; if it wasn't for that, you'd call differentiation an art as well. I.e., we think of "nice" functions as those built up by compositions from some basic stock of primitive functions we understand well; the chain rule allows us to take our knowledge of the derivatives of the primitive functions and easily build up from this knowledge of the derivatives of all other "nice" functions, in this sense. Integration has no such chain rule, so it's hard | |
| May 29, 2011 at 20:07 | comment | added | vonjd | @Ryan: That it is easy and doable one way and hard or sometimes not even doable the other way round. I am trying to understand the structural (deep) reason for that asymmetry. When you say isn't clear what are the alternative interpretations or what is missing? | |
| May 29, 2011 at 20:01 | comment | added | Ryan Budney | Your question isn't clear. What's this "other" assymmetry you're referring to? | |
| May 29, 2011 at 19:54 | comment | added | vonjd | @Qiaochu: I understand that but my question is why this asymmetry is the case (the underlying reason) with differentiation and integration. | |
| May 29, 2011 at 19:43 | answer | added | harecare | timeline score: 16 | |
| May 29, 2011 at 19:12 | comment | added | The Mathemagician | Differentiation is a "linearizing" limit process: it results in an approximation to the original function that is a linear map. Integration does not necessarily result in a linear map from the original function-quite the opposite most of the time. That's my 2 cents on the question. | |
| May 29, 2011 at 17:30 | answer | added | Todd Trimble | timeline score: 146 | |
| May 29, 2011 at 17:24 | comment | added | Eric Naslund | @Vonjd: A similar question was asked on MSE. This thread may be of interest: math.stackexchange.com/questions/20578/… | |
| May 29, 2011 at 17:00 | comment | added | Qiaochu Yuan | The fundamental theorem of calculus tells you that they're inverses. That's not necessarily a symmetry. There are plenty of examples of functions that are easy to compute whose inverses are hard to compute: en.wikipedia.org/wiki/One-way_function | |
| May 29, 2011 at 16:56 | comment | added | vonjd | @Suvrit: Nice one - ROFL :-) | |
| May 29, 2011 at 16:52 | comment | added | C.S. | Because differentiation is easier than integration. | |
| May 29, 2011 at 16:49 | comment | added | Suvrit | differentiation increases entropy, integration reduces it, so physics answers the asymmetry (just kidding). | |
| May 29, 2011 at 16:44 | history | asked | vonjd | CC BY-SA 3.0 |