Re: RE: Re: Dot Product in Cylindrical Coordinates

• To: mathgroup at smc.vnet.net
• Subject: [mg69351] Re: [mg69301] RE: [mg69276] Re: Dot Product in Cylindrical Coordinates
• From: Pratik Desai <pratikd at wolfram.com>
• Date: Thu, 7 Sep 2006 23:58:49 -0400 (EDT)
• References: <200609060828.EAA12847@smc.vnet.net>

```David Park wrote:
> Paul,
>
> I want to expand a little more on my previous posting.
>
> The point is that the VectorAnalysis does expect that the components of a
> vector are given in the orthornormal frame in Curl and Divergence. So it is
> a little quixotic to switch the context when using DotProduct and
> CrossProduct.
>
> Needs["Calculus`VectorAnalysis`"]
> SetCoordinates[Cylindrical[\[Rho], \[Phi], z]];
>
> Here is a theorem from vector calculus:
>
> div[g x f] == curl[g].f - curl[f].g
>
> Which dot product is meant? If we use the ordinary Dot product the theorem
> is true.
>
> Div[Cross[{g\[Rho][\[Rho], \[Phi], \[Theta]],
>         g\[Phi][\[Rho], \[Phi], \[Theta]],
>         gz[\[Rho], \[Phi], \[Theta]]}, {f\[Rho][\[Rho], \[Phi], \[Theta]],
>         f\[Phi][\[Rho], \[Phi], \[Theta]], fz[\[Rho], \[Phi], \[Theta]]}]]
> ==
>   DotProduct[
>       Curl[{g\[Rho][\[Rho], \[Phi], \[Theta]],
>           g\[Phi][\[Rho], \[Phi], \[Theta]],
>           gz[\[Rho], \[Phi], \[Theta]]}], {f\[Rho][\[Rho], \[Phi],
> \[Theta]],
>         f\[Phi][\[Rho], \[Phi], \[Theta]], fz[\[Rho], \[Phi], \[Theta]]}] -
>     DotProduct[
>       Curl[{f\[Rho][\[Rho], \[Phi], \[Theta]],
>           f\[Phi][\[Rho], \[Phi], \[Theta]],
>           fz[\[Rho], \[Phi], \[Theta]]}], {g\[Rho][\[Rho], \[Phi],
> \[Theta]],
>         g\[Phi][\[Rho], \[Phi], \[Theta]], gz[\[Rho], \[Phi], \[Theta]]}];
> % // Simplify
> True
>
> If instead we use the VectorAnalysis DotProduct the theorem is not true.
>
> Div[Cross[{g\[Rho][\[Rho], \[Phi], \[Theta]],
>           g\[Phi][\[Rho], \[Phi], \[Theta]],
>           gz[\[Rho], \[Phi], \[Theta]]}, {f\[Rho][\[Rho], \[Phi], \[Theta]],
>           f\[Phi][\[Rho], \[Phi], \[Theta]],
>           fz[\[Rho], \[Phi], \[Theta]]}]] ==
>     DotProduct[
>         Curl[{g\[Rho][\[Rho], \[Phi], \[Theta]],
>             g\[Phi][\[Rho], \[Phi], \[Theta]],
>             gz[\[Rho], \[Phi], \[Theta]]}], {f\[Rho][\[Rho], \[Phi], \
> \[Theta]], f\[Phi][\[Rho], \[Phi], \[Theta]], fz[\[Rho], \[Phi],
> \[Theta]]}] -
>        DotProduct[
>         Curl[{f\[Rho][\[Rho], \[Phi], \[Theta]],
>             f\[Phi][\[Rho], \[Phi], \[Theta]],
>             fz[\[Rho], \[Phi], \[Theta]]}], {g\[Rho][\[Rho], \[Phi], \
> \[Theta]], g\[Phi][\[Rho], \[Phi], \[Theta]], gz[\[Rho], \[Phi],
> \[Theta]]}];
> % // Simplify
> (output omitted)
>
> This is certainly a 'feature' and I think it is an error in design. The user
> should at least be warned about the change in context and meaning. The
> documentation uses the phrase 'in the default coordinate system' in both
> DotProduct and in Curl so one could become easily confused.
>
> David Park
>
> From: Paul Abbott [mailto:paul at physics.uwa.edu.au]
To: mathgroup at smc.vnet.net
>
>
> In article <edadqd\$pgc\$1 at smc.vnet.net>,
>  Sergio Miguel Terrazas Porras <sterraza at uacj.mx> wrote:
>
>
>> When I calculate the dot product of vectors {1,Pi/4,0} and {2,0,1} in
>> Cylindrical Coordinates Mathematica 5.1 returns the result Sqrt[2], when
>>
> the
>
>> result should be 2.
>>
>
> Notwithstanding several of the other responses, the result _is_ Sqrt[2].
> When you write {1,Pi/4,0}, surely you mean
>
>   {rho, phi, z} == {1, Pi/4, 0}
>
> and _not_ that
>
>   {x, y, z} == {1, Pi/4, 0} ?
>
>
>   Needs["Calculus`VectorAnalysis`"]
>
> and selecting Cylindrical coordinates,
>
>   SetCoordinates[Cylindrical];
>
> then in cartesian coordinates, this point is
>
>   p1 = CoordinatesToCartesian[{1, Pi/4, 0}]
>
>   {1/Sqrt[2], 1/Sqrt[2], 0}
>
> Similarly,
>
>   p2 = CoordinatesToCartesian[{2, 0, 1}]
>
>   {2, 0, 1}
>
> Hence the dot product of the coordinate vectors (relative to the origin
> {0,0,0}), computed in cartesian coordinates, is
>
>   p1 . p2
>
>   Sqrt[2]
>
> This is the same result that you got, presumably using,
>
>   DotProduct[ {1, Pi/4, 0}, {2, 0, 1} ]
>
>   Sqrt[2]
>
> Of course, if you really mean
>
>   {x, y, z} == {1, Pi/4, 0}
>
> then there is no need to load Calculus`VectorAnalysis`: the dot product
> is just
>
>   {1, Pi/4, 0} . {2, 0, 1}
>
>   2
>
> Note that Dot is _not_ modified when this package is loaded so
> Jean-Marc's response,
>
>   Needs["Calculus`VectorAnalysis`"]
>   SetCoordinates[Cylindrical];
>   {1, Pi/4, 0} . {2, 0, 1}
>
> is bogus -- the first two lines have no effect on the third.
>
> Modifying Andrzej's code, we have
>
>  Simplify[(JacobianMatrix[] . p1) . (JacobianMatrix[] . p2)]
>
>  Sqrt[2]
>
> Cheers,
> Paul
>
> _______________________________________________________________________
> Paul Abbott                                      Phone:  61 8 6488 2734
> School of Physics, M013                            Fax: +61 8 6488 1014
> The University of Western Australia         (CRICOS Provider No 00126G)
> AUSTRALIA                               http://physics.uwa.edu.au/~paul
>
>
I think it is quite imperative to note here that

"There are often conflicting definitions of a particular coordinate
system in the literature. When you use a coordinate system with this
package, you should look at the definition given below to make sure it
is what you want." --Mathematica Documentation

So for cylindrical coordinate system one must define the system as:

g = {g?[r, theta, z], g?[r, theta, z], gz[r, theta, z]}
f = {f?[r, theta, z], f?[r, theta, z], fz[r, theta, z]}
g?[r_, theta_, z_] = r
g?[r_, theta_, z_] = theta
gz[r_, theta_, z_] = z
f?[r_, theta_, z_] = r^2
f?[r_, theta_, z_] = theta
fz[r_, theta_, z_] = Cos[z]

Then everything works fine:

In[20]:=
Div[CrossProduct[g,f]]===DotProduct[Curl[g],f]-DotProduct[Curl[f],g]

Out[20]=
True

Hope this helps

Pratik

```

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