Timeline for What is the geometric meaning of the third derivative of a function at a point? [closed]
Current License: CC BY-SA 3.0
40 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Aug 27, 2014 at 17:45 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
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| Jul 20, 2013 at 19:17 | history | edited | Gil Kalai | CC BY-SA 3.0 |
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| Dec 20, 2010 at 11:30 | history | closed |
Andy Putman Theo Johnson-Freyd Charles Siegel Harry Gindi Andrew Stacey |
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| Dec 20, 2010 at 7:16 | history | edited | Gil Kalai | CC BY-SA 2.5 |
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| Dec 20, 2010 at 2:31 | comment | added | drbobmeister | @Gil Kalai: Well, now that you've edited it, it seems fine to me. Some may say there is not enough "motivation" but some questions are self-motivating, i.e. they don't require lots of explanation. This one seems to me to be in that category | |
| Dec 19, 2010 at 23:45 | comment | added | Gil Kalai | Maybe the original motivation was to use the word "of" three times. | |
| Dec 19, 2010 at 21:38 | comment | added | Pete L. Clark | As Gil Kalai says, the question has been reasked at math.stackexchange.com/questions/14841/… and has already received five answers. I am a bit perplexed as to why people think it is a good thing to have the same question open on both sites. If you feel this way, would you please explain yourself on the meta thread that has been opened on this question? | |
| Dec 19, 2010 at 20:27 | history | reopened |
Dick Palais algori KConrad Gil Kalai Georges Elencwajg |
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| Dec 19, 2010 at 9:58 | comment | added | Gil Kalai | I asked the question over the sister site math.stackexchange.com/questions/14841/… My feeling is that (edited of course) this is a good MO question because it is certainly a question that is on the minds of many mathematicians in the context of teaching and also in some research contexts, and I am not aware of very good answers. | |
| Dec 19, 2010 at 8:14 | comment | added | drbobmeister | The problem with re-writing a better but thematically similar question is that everybody here has an answer, but the question has been taken away! How to turn an answer into a question, that's my problem! | |
| Dec 19, 2010 at 7:40 | comment | added | drbobmeister | Furthermore, it is true that the original post had severe grammatical errors. Nevertheless, in essence the question remains as first stated. Like many simple questions in mathematics, it opens the way to some much deeper ideas. | |
| Dec 19, 2010 at 6:39 | history | edited | Gil Kalai |
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| Dec 19, 2010 at 6:07 | comment | added | drbobmeister | BTW, has anyone heard from AJAY, our OP? He doesn't seem to have stuck around for the discussion! | |
| Dec 19, 2010 at 5:25 | history | edited | Gil Kalai | CC BY-SA 2.5 |
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| Dec 19, 2010 at 3:55 | comment | added | sleepless in beantown | also, +1 Andy Putman. | |
| Dec 19, 2010 at 3:51 | comment | added | sleepless in beantown | @Theo Johnson-Freyd, ++ two thumbs up, Siskel and Ebert style. Note that I'm happy with my answer in the comments. IF the original poster improves and clarifies, it might be worth considering reopening it, but I doubt it. I think that people jump the gun too quickly on closing questions without seeing the subtleties that might be involved. What's wrong with leaving it open for a while? Also, I put a third-up vote on it. Someone down-voted it without leaving any sort of explanation. I agree that this question is malformed. If the OP fixes the style and content, then I'll rethink it. :) | |
| Dec 19, 2010 at 3:40 | comment | added | Theo Johnson-Freyd | Given all the discussion here, I think that there's probably a related question that deserves to be opened. But I do not support reopening this question as written: the question is poorly written, unclear, and requires both editing both at the levels of style and content. | |
| Dec 19, 2010 at 1:26 | comment | added | Andy Putman | Though this question does have a reasonable answer, it seems to me that it is not at the level of MO. It would be more appropriate for math.stackexchange.com. | |
| Dec 18, 2010 at 23:22 | comment | added | sleepless in beantown | @Tom LaGatta, My personal opinion is that it is the question-writer's duty/obligation to improve the question. There's not enough time to answer the questions that are asked, let alone answer the questions not asked. "I was just adding my two cents to this discussion, which I would vote to reopen if the questioner edited/clarified the question to add context" The original poster would need to clarify what they intend by "geometrical meaning" and "curvature", as different meanings lead to different answers. | |
| Dec 18, 2010 at 23:17 | comment | added | Tom LaGatta | drbobmeister, sleepless in beantown and Dick Palais: You all are right that there are plenty of great answers to this poorly phrased question. If one of you reformulates it in the form of a good question, then I'm sure everybody will support reopening it. | |
| Dec 18, 2010 at 23:10 | comment | added | drbobmeister | Quite frankly I fail to see what's "unreal" about this question. | |
| Dec 18, 2010 at 23:10 | comment | added | drbobmeister | Finally (again), I wiki'ed "curvature" when I wrote my previous comment, and at least in that article I didn't see the whole stuck as hinted at here, though I didn't review the entry thoroughly--this time. "There's more things in heavan and earth, Horatio, than you'll ever find in your wiki article."--Bill Jerkspeare. | |
| Dec 18, 2010 at 23:07 | comment | added | drbobmeister | @sleepless--I by no means fell we're at odds on this subject. I tried to write with severe brevity since I'm working in "comment space", not "answer space". (BTW, can closed questions be answered? I was under the impression the answer is "no".) Anyway, being cognizant of the geometric sense in using "per length" than "per parameter increment" I mentioned a more differential geometric approach, which I believe clarifies the issues. | |
| Dec 18, 2010 at 22:52 | comment | added | sleepless in beantown | @drbobmeister, I wasn't disagreeing with you. I was just trying to be "rigorous" and defining specifically what I meant by curvature, as curvature for a geometric object ought be defined by change in unit distance over the geometric object. I wasn't trying to say you misdefined curvature; I noted your specificity in your answer. I was just adding my two cents to this discussion, which I would vote to reopen if the questioner edited/clarified the question to add context. | |
| Dec 18, 2010 at 22:38 | comment | added | drbobmeister | More, to beat the 500 character limit: Finally, I think a more geometric view of physics (say a la Einstein) sees space and time on a more equal footing. Nath'less (as Chaucer would say), I basically agree with sleepless. We're in the same chapter, if not on the same page. Again, there's more here than meets the eye, as Dick and sleepless have noticed. | |
| Dec 18, 2010 at 22:37 | comment | added | drbobmeister | Though I basically agree with sleepless, please note that in my comment I said that $d^{3}y/dx^{3}$ is related to curvature, not that it is curvature! Also, the outworking of these ideas in a coordinate-free differential geometric notation will give (I betcha!) rate of change of curvature with arc length (the natural parameter for a curve; then the tangent vector is of unit length). | |
| Dec 18, 2010 at 22:36 | history | edited | sleepless in beantown |
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| Dec 18, 2010 at 22:35 | history | edited | sleepless in beantown |
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| Dec 18, 2010 at 22:30 | comment | added | sleepless in beantown | Curvature geometrically over a shape/function should be defined as the change in direction of the function per unit distance travelled along the shape/function, not by the change in slope per unit distance in the x-direction. My example for a circle should clarify that: $x=r cos(t), y=r sin(t)$ is a curve with constant curvature and thus change in curvature geometrically is $0$. A circle as $y = \pm \sqrt{r^2-x^2}$ will have slope per change in x varying over x, and thus curvature and change in curvature will vary over x. Geometrically, curvature is (ought to be?) defined differently. :) | |
| Dec 18, 2010 at 22:26 | comment | added | sleepless in beantown | Since I've pretty much put my version of an answer in the comments, it should be obvious that I agree with @Dick-Palais that this question deserves an answer. Of course, the question could have been better posed with some more context around it, and a modicum of effort on the questioner's part could have added some substance to the question. But note that in my answer in these comments, I point out a subtlety in the definition of curvature geometrically vs. physically. In physics, if $f$ is a function of $t$=time, then "jerk" is the correct answer. Geometrically, this is not the case. | |
| Dec 18, 2010 at 22:22 | comment | added | sleepless in beantown | Change in curvature might better be defined for parametric curves: for example, define a circle as a curve. If it's not defined parametrically, then the change in curvature vs. the change in $x$ varies at different points over the circle; but if the curvature is defined over a specific parametric equation of the points of the circle, then the curvature is constant, and the change in curvature is zero, as the curvature of the circle is constant over all points on the circle. Thus the way the curve is defined or parametrized is a significant component in the definition of curvature and f'''. | |
| Dec 18, 2010 at 22:17 | comment | added | sleepless in beantown | Richard Kent's answer is correct for the physical interpretation: if f(x)=position at time x, f'(x)=velocity=v=dx/dt, f''(x)=acceleration=dv/dt, f'''(x)=jerk=$\delta a / \delta t$, change in acceleration over time. The geometrical interpretation is as drbobmeister says f'''(x)=rate of change of curvature as x changes, f''(x)=curvature=rate of change of slope vs. dx, f'(x)=slope=change if f(x) per change in x. So it's not really the change in curvature of a curve, but the change in curvature per change in $x$. Change in curvature might better be defined for parametric curves. | |
| Dec 18, 2010 at 21:49 | comment | added | drbobmeister | If I could vote on such matters, I would vote to re-open as well. | |
| Dec 18, 2010 at 21:48 | comment | added | drbobmeister | Since the second derivative basically relates to curvature, viz. for a simple "curve" $y = f(x)$ we have $k = (d^{2}y/dx^{2})/(1 + (dy/dx)^{2})^{3/2}$, the third derivative relates to the rate of change of curvature. This could seemingly be worked out in coordinate-independent terms using tangent and normal vectors, etc. (Think Frenet-Serre formulae.) My guess is that, in $R^{3}$ and higher (dimensions), torsion etc. enters in. So it seems there is some geometry here after all! | |
| Dec 18, 2010 at 21:29 | comment | added | José Figueroa-O'Farrill | Dick: I think that questions which can be answered by a wikipedia link are not deemed to be appropriate on MO, hence the "not a real question" reason for closure. Your comment (or even Richard's) should be sufficient to answer the question. | |
| Dec 18, 2010 at 20:38 | comment | added | Dick Palais | I'm surprised at the claim it is not a "real" question. It is! For example see here. en.wikipedia.org/wiki/Jerk_%28physics%29 I feel this should be re-opened. | |
| Dec 18, 2010 at 19:39 | history | closed |
Bill Johnson Andrés E. Caicedo Felipe Voloch Angelo Qiaochu Yuan |
not a real question | |
| Dec 18, 2010 at 19:14 | comment | added | Autumn Kent | It is the jerk! | |
| Dec 18, 2010 at 19:03 | history | asked | AJAY | CC BY-SA 2.5 |