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Re: Re: cross-product in cylindrical problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51681] Re: [mg51646] Re: cross-product in cylindrical problem
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 29 Oct 2004 03:38:59 -0400 (EDT)
  • References: <200410100952.FAA25275@smc.vnet.net> <3B366122-1AC0-11D9-BEB1-000A95B4967A@mimuw.edu.pl> <ckd5nu$50d$1@smc.vnet.net> <200410280343.XAA09708@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Actually you should be able to read it if you use Unicode as your 
default encoding format for messages or if your mail program allows you 
to choose the encoding that should be used for displaying a message. 
Most modern mailers can do this and Unicode is used by all fairly 
modern OSs known to me. (Since Apple's Mail uses Unicode by default I 
keep forgetting that it can cause problems when copying and pasting 
plain text from Mathematica.)
Anyway, here is the same thing in a less elegant but universally 
readable form:

SetCoordinates[Cylindrical[\[Rho], \[Phi], z]];

Andrzej

On 28 Oct 2004, at 12:43, news wrote:

> hello Andrzej,
> Thank you for your reply, but I can't read your fonts, can you explain 
> what
> the symobls are suppposed to be in: SetCoordinates[Cylindrical[Ï?, Ï., 
> z]];
>
> thanks very much!
>
> "Andrzej Kozlowski" <akoz at mimuw.edu.pl> wrote in message
> news:ckd5nu$50d$1 at smc.vnet.net...
>> I forgot to include the line:
>>
>> SetCoordinates[Cylindrical[Ï?, Ï., z]];
>>
>> Without that JacobianMatrix[] should be replaced by
>> JacobianMatrix[Cylindrical[Ï?, Ï., z]].
>>
>> Andrzej
>>
>>
>> On 10 Oct 2004, at 22:28, Andrzej Kozlowski wrote:
>>
>>> On 10 Oct 2004, at 18:52, news wrote:
>>>>
>>>> I'm really puzzled by this behavior of Mathematica, I have two
>>>> vectors in
>>>> cylindrical coordinates and would like to take their cross-product 
>>>> in
>>>> cylindrical, but it seems to give me incorrect answer, see below:
>>>>
>>>> define parametric path {r,phi,z}
>>>>
>>>> In[110]:=
>>>> f[\[Rho]_, \[Phi]_] = {\[Rho], \[Phi], 0}
>>>> Out[110]=
>>>> {\[Rho], \[Phi], 0}
>>>>
>>>> take derivates of path w.r.t. r then w.r.t phi, get {1,0,0}, and
>>>> {0,1,0}
>>>>
>>>> In[113]:=
>>>> v1 = D[f[\[Rho], \[Phi]], \[Rho]]
>>>> v2 = D[f[\[Rho], \[Phi]], \[Phi]]
>>>> Out[113]=
>>>> {1, 0, 0}
>>>> Out[114]=
>>>> {0, 1, 0}
>>>>
>>>> then cross them in cylindrical coords, and should get {0,0,1}, but
>>>> instead
>>>> get wrong answer below
>>>>
>>>> In[117]:=
>>>> n = CrossProduct[v1, v2, Cylindrical[\[Rho], \[Phi], z]] //
>>>> FullSimplify
>>>> Out[117]=
>>>> {0, 0, 0}
>>>>
>>>> As you can see, when I cross {1,0,0} with {0,1,0} in cylindrical
>>>> coords, I
>>>> get {0,0,0}, when I should be getting {0,0,1}.
>>>>
>>>> Can anyone help?
>>>>
>>>>
>>> The problem is with the meaning of "two vectors in cylindrical
>>> coordinates". Even if your default coordinate system is cylindrical,
>>> Mathematica still represents tangent vectors using Cartesian
>>> coordinates. In other words, the meaning of a vector {a,b,c} in
>>> cylindrical coordinates is the vector given by:
>>>
>>> << Calculus`VectorAnalysis`
>>>
>>> CoordinatesToCartesian[{a,b,c},Cylindrical]
>>>
>>> Thus, the vectors {1,0,0} and {0,1,0} are turned to:
>>>
>>>
>>> CoordinatesToCartesian[{1, 0, 0}, Cylindrical]
>>>
>>> {1, 0, 0}
>>>
>>> and
>>> CoordinatesToCartesian[{0,1,0},Cylindrical]
>>>
>>> {0,0,0}
>>>
>>> So you get the zero vector (since {0,1,0} in cylindrical coordinates
>>> represents just the origin) and hence the result you get.
>>> I am not hundred percent sure what you meant. But I suppose thatt by
>>> {1,0,0} and {0,1,0} in cylindrical coordinates you meant the basic
>>> vector fields in the tangent bundle of R^3 (in other words vectors
>>> changing from point to point). Presumably (I am guessing) the vector
>>> (field) {1,0,0} is
>>>
>>> JacobianMatrix[] . {1, 0, 0}
>>>
>>> {Cos[Ï.], Sin[Ï.], 0}
>>>
>>>  and the vector (field) {0,1,0} is
>>>
>>> JacobianMatrix[] . {0, 1, 0}
>>>
>>> {(-Ï?)*Sin[Ï.], Ï?*Cos[Ï.], 0}
>>> in Cartesian coordinates.
>>>
>>> In that case
>>>
>>> Simplify[Cross[JacobianMatrix[] . {1, 0, 0}, JacobianMatrix[] . {0, 
>>> 1,
>>> 0}]]
>>>
>>> {0, 0, Ï?}
>>>
>>> which is the vector field sometimes denoted by {0,0,1} in cylindrical
>>> coordinates.
>>>
>>> Andrzej Kozlowski
>>> Chiba, Japan
>>> http://www.akikoz.net/~andrzej/
>>> http://www.mimuw.edu.pl/~akoz/
>>>
>>>
>>
>>
>


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