       Re: Vector Fields

• To: mathgroup at smc.vnet.net
• Subject: [mg14716] Re: Vector Fields
• From: Rainer.Bassus at t-online.de (Rainer Bassus)
• Date: Tue, 10 Nov 1998 01:21:06 -0500
• References: <7258kj\$8bi@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```David Farrelly wrote:<7258kj\$8bi at smc.vnet.net>...
>
>I have a couple of questions that I'd appreciate some suggestions on;
>
>(i) I can't get PlotVectorField to work on Mathematica running
>    on a PC. Basically I can't figure out what packages
>    to load based on what the manual says.
>
You can plot a vector field in Cartesian coordinates with the following
steps:

A = ex Ax + ey Ay + ez Az  **This is the Vector field**

ex, ey, ez  **are the unit vectors**

Ax = fx(x,y,z), Ay = fy(x,y,z), Az = fz(x,y,z) Ax, Ay, Az **are three
different functions of x,y,z**

A very simple vector field is:
Ax = 0, Ay = 0, Az = 1 ** this vector field has no function of x,y,z but
an constant value of 1 in z-direction ***

The plot with Mathematica can be done with: A = {0,0,1};
<<Graphics`PlotField3D`;
PlotVectorField3D[ A ,{x,-1,1},{y,-1,1},{z,-1,1}]

>(ii) Does anyone know how to plot a vector field on a sphere.
>     What I'd like to do is to represent fluid flow using colors
>     to represent the fluid density and arrows to represent the
>     vector field. This is done on the surface of a sphere (it is
>     a quantum mechanical probability current rather than a real
>     fluid). If anyone knows of non Mathematica ways of doing
>     this I'd also be interested.
>
Mathematica doesn't support the statement SpericalPlotVectorField3D
so you have to convert the Spherical vector field into a Cartesian vector
field and choose than the statement PlotVectorField3D.

The Spherical vector field:

A = eph Aph + eth Ath + er Ar   ( eph, eth, er are the units vectors
(bold))

Every part of this vector field in normally a function of phi, theta and
r:

Aph = f1(ph,th,r)       (something like that: Aph = r^2Sin[th] Cos[ph])
Ath  = f2(ph,th,r)       (something like that: Ath =r^7Sin[th]^2
Cos[ph]^3) Ar    = f3(ph,th,r)

as sample we choose a very simple field:

Aph = f1(ph,th,r) = r        (no function of phi,th. But a function of r
) Ath  = f2(ph,th,r) = 0       (no function of phi,th, r  and no
constant value)
Ar    = f3(ph,th,r) = 0       (no function of phi,th, r  and no constant
value)

Asph = {Ar,Ath,Aph}         ( That is our entire field)

For the PlotVectorField3D-command you need a x,y,z-dependent. So you
have to replace every r, sin[theta] ,sin[phi] by:

r = Sqrt[ x^2 + y^2 + z^2];
sinph = y/Sqrt[x^2 + y^2];
cosph = x/Sqrt[x^2 + y^2];
sinth = Sqrt[(x^2+y^2)/(x^2+y^2+z^2)]; costh = Sqrt[ z^2/(x^2+y^2+z^2)];

The sample vector field results in:

Aph = f1(x,y,z) = Sqrt[ x^2 + y^2 + z^2] Ath  = f2(x,y,z) = 0
Ar    = f3(x,y,z) = 0          (This is a Spherical vector field) Asph =
{ 0 , 0 , Sqrt[ x^2 + y^2 + z^2] }         ( That is our entire field)

And now the transformation into cartasian coordinates. The standard
transformation is (I found this in "Bronstein Semendjajew - Taschenbuch
der Mathematik"):

Acat =Simplify[{
Asph[] cosph sinth - Asph[] sinph + Asph[] cosph costh,
Asph[] sinph sinth + Asph[] cosph + Asph[] sinph costh,
Asph[]       costh                                  - Asph[]
sinth }];

Now the plot with Mathematica:

<<Graphics`PlotField3D`;
PlotVectorField3D[ Acat,{x,-1,1},{y,-1,1},{z,-1,1}]

Rainer Bassus

```

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