Re: Bug in D (or Derivative) ( Was: Re: DSolve bug)

*To*: mathgroup at smc.vnet.net*Subject*: [mg97941] Re: [mg97919] Bug in D (or Derivative) ( Was: Re: DSolve bug)*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 26 Mar 2009 05:21:28 -0500 (EST)*References*: <612696379.19321237704569790.JavaMail.root@mmm.inp.demokritos.gr> <200903251046.FAA24696@smc.vnet.net> <8DF2CBD8-6542-461B-9A52-CDE33E721DA5@mimuw.edu.pl>

On 25 Mar 2009, at 14:40, Andrzej Kozlowski wrote: > > On 25 Mar 2009, at 11:46, schochet123 at gmail.com wrote: > >> >> >> The source of this bug seems to be the following bug in in D (or >> Derivative): According to Mathematica, D[u',u] yields u''. However, >> the correct answer is u''/u' >> (Homework for Calculus 1: use the chain rule to prove this.) > > Not really. According to Mathematica's notion of derivative, D[u',u] > ought to be 0, since u' does not involve u explicitly. Actually, I was wrong here since, of course, u' is Derivative[1][u] and this does involve u explicitly. Of course in this case D[Derivative[1][u],u] does not make any mathematical sense (unless we wanted this to mean the derivative in some functional sense, but then what Mathematica does with it makes no sense anyway). This: D[derivative[1][u], u] Derivative[1][derivative[1]][u] does make sense, as long as derivative is undefined, but this: % /. derivative -> Derivative Derivative[2][u] doesn't. Still, for Mathematica to be consistent and still make mathematical sense, D[u',u] ought to return 0, as it in the example below. Andrzej Kozlowski > In fact, > > D[u'', u'] > 0 > > which, of course, is much the same thing. The only way I know to get > the answer you have in mind with Mathematica is: > > Dt[Derivative[1][u][x], u[x]] > Derivative[2][u][x]*Dt[x, u[x]] > > >> >> >> The way Mathematica obtains its incorrect result is quite amusing, As >> Trace shows, Mathematica evaluates D[u',u] by noting that the >> FullForm >> of u' is Derivative[1][u], whose derivative with respect to u is (of >> course ;) ) Derivative[1]'[u]. >> Mathematica then decides that Derivative[1]' is Derivative[2], >> yielding Derivative[2][u], i.e., u''. >> > > Indeed, this is right, and has been mentioned on this forum several > times in the past (at least twice by me). > See, for example, > > http://forums.wolfram.com/mathgroup/archive/2008/Jul/msg00801.html > > > Andrzej Kozlowski > > >

**References**:**Bug in D (or Derivative) ( Was: Re: DSolve bug)***From:*schochet123@gmail.com