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Re: cross-product in cylindrical problem

• To: mathgroup at smc.vnet.net
• Subject: [mg51263] Re: [mg51258] cross-product in cylindrical problem
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 11 Oct 2004 01:25:22 -0400 (EDT)
• References: <200410100952.FAA25275@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

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/

• References:
  • cross-product in cylindrical problem
    • From: "news" <anonym@bamboo.com>