**Density flow of a
vector field along the roman surface of Steiner**

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**Prof.
E.Musso, PhD.**

*Dipartimento di
Matematica Pura ed Applicata*

*Università de
L’Aquila- L’Aquila-Italy.*

*e-mail:
musso@univaq.it*

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**Abstract**

In this paper we consider the density flow of a
vector field along the Steiner’s Roman surface. The components of the vector
field are Wierstrass’s elliptic functions. Symbolic and numerical computations,
as well the visualization, are performed with the software Mathematica 5.1.

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**Denisity flow
of a vector field along the roman surface of Steiner**

**Fig.
1 : the roman surface of Steiner.**

The roman surface
is defined by the parametric equations

(1)
x(u,v)=cos(u)^{2}cos(v)sin(v),

(2)
y(u,v)=cos(u)sin(u)cos(v),

(3)
z(u,v)=cos(u)sin(u)sin(v).

**Fig. 2 : densità del
flusso del campo vettoriale W**

This
picture represents the density of the flow of the vector field W with components

(4)
W^{1}(u,v) = Re[σ(u+iv, ω_{1}+
iω_{2})],

(5)
W^{2}(u,v) =
Im[*P*(u+iv, ω_{1}+ iω_{2})],

(6)
W^{3}(u,v) =
Im[σ(u+iv, ω_{1}+ iω_{2})],

where
P and σ denote the Weierstrass
elliptic functions with fundamental periods ω_{1}=1 and
ω_{2}=2. The
density through the surface is given by

(7)
ρ =
N∙W (g_{11} g_{22} - g_{12}^{2})^{1/2},

where
the functions g_{ij} , i,j=1,2, are the coefficients of the first
fundamental form of the surface and N denotes the Gauss map. The visualization
of the density is obtained by the means of the following “color function”

(8)
CF(u,v) =
Hue(-1/2(1+(1+ ρ)(1+
ρ^{2})^{-1}),1,1)

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**The program**

The program has been written with the software Matematica 5.1.

**Fig. 4: the program**