Help for a "Surface and Flux integrals" program

[salvomic]

hi,
I need some help to implement for the Prime a simple CAS programs to calculate Surface Integrals and Flux Integrals (Surface integrals with vector fields)

Surface integral:
\[ \int _\sigma f dS = \iint_{A}f(\sigma _1(u,v),\sigma _2(u,v),\sigma _3(u,v))\sqrt{I_1^2+I_2^2+I_3^2}dudv \]

Flux integral:
\[
\int _\sigma F\cdot \mathbf{n} \, dS = \iint_{\sigma }(f_1(\sigma _1(u, v),\sigma _2(u, v),\sigma _3(u, v))I_1)+f_2(\sigma _1(u, v),\sigma _2(u, v),\sigma _3(u, v))I_2+f_3(\sigma _1(u, v),\sigma _2(u, v),\sigma _3(u, v))I_3) dudv
\]
(where \( F = (f_1, f_2, f_3) \))

Where also:
\[
I_1(u,v) = det\begin{pmatrix}
\frac{\partial \sigma _2}{\partial u}(u,v) & \frac{\partial \sigma _2}{\partial v}(u,v)\\
\frac{\partial \sigma _3}{\partial u}(u,v) & \frac{\partial \sigma _3}{\partial v}(u,v)\\
\end{pmatrix}
\, ;
I_2(u,v) = det\begin{pmatrix}
\frac{\partial \sigma _3}{\partial u}(u,v) & \frac{\partial \sigma _3}{\partial v}(u,v)\\
\frac{\partial \sigma _1}{\partial u}(u,v) & \frac{\partial \sigma _1}{\partial v}(u,v)\\
\end{pmatrix}
\, ;
I_3(u,v) = det\begin{pmatrix}
\frac{\partial \sigma _1}{\partial u}(u,v) & \frac{\partial \sigma _1}{\partial v}(u,v)\\
\frac{\partial \sigma _2}{\partial u}(u,v) & \frac{\partial \sigma _2}{\partial v}(u,v)\\
\end{pmatrix}

\]

Which are parts of: \( I = (I_1, I_2, I_3) \)

I would like to start (or follow) from this program in the Prime Software Library that has a simple syntax (for linear and curvilinear integrals):
INPUT 4 parameters: 1. a function (scalar / vectorial), 2. parametrisation of a curve, 3. lower bound, 4. upper bound;
and a control for arguments (2 or 3) and for the case there is no input (and then the program show a little help)...

I'd think to extend these concepts (from (curvi)linear integrals to surface and flux), using a parametrisation of the surface like "u, v, σ(u, v)"...
\[ \sigma (u, v) : \left\{\begin{matrix}
x_{1}=\sigma_{1}(u, v)) \\
x_{2}=\sigma_{2}(u, v)) \\
x_{3}=\sigma_{3}(u, v))
\end{matrix}\right.
\,\, , (u,v) \in A \subset \mathbb {R}^2
\]
But I've no clear idea at the moment, ehm :-)

Thank you in advance!
Salvo

EDIT: edited typo in laTeX formulas (thanks Alex!)...