Surface and flux integrals

[Arno K]

Inspired by Salvos post:

(11-03-2017 05:53 PM)salvomic Wrote:  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)"...
\[ \sigma (u, v) : \left\{\begin{matrix}
x_{1}=\sigma_{1}(u, v)) \\
x_{2}=\sigma_{2}(u, v)) \\
x_{3}=\sigma_{3}(u, v))
\end{matrix}\right.
\,\, , (u,v) \in A \subset \mathbb {R}^2
\]

I wrote that little cas program for the Prime:

PHP Code:

#cas
sfint(f,g,uvals,vvals):=
BEGIN
local gg, u, v;
 purge(u);
 purge(v);// as long as no other solution this has to be put here to avoid the occurence of u and v as cas-vars afterwards
//  Help;
//  Usage:sfint(f(x,y,z),[φ1(u,v),φ2(u,v),φ3(u,v)],[ulow,uhigh],[vlow,vhigh])
//  f can be a vector, too, i.e.:[x*y*z,x+y,x^2+z^3]


f:=subst(f,[x,y,z]=g);// perform the substitution in f
gg:=transpose(grad(g,[u,v]));//compute the jacobian
gg:=cross(col(gg,1),col(gg,2));
IF type(f)==DOM_LIST THEN// in case a vector-function is entered
f:=DOT(f,gg);
ELSE
f:=f*ABS(gg);//f is a scalar
END;
f:=int(f,u,uvals[1],uvals[2])
  return int(f,v,vvals[1],vvals[2]);//return the double integral
END;
#end 


Input can either be \((f(x,y,z),[\varphi 1(u,v),\varphi 2(u,v),\varphi 3(u,v)],[\mathrm{ulow},\mathrm{uhigh}],[\mathrm{vlow},\mathrm{vhigh}])\)
or f can be a vector to compute the flux through the surface.
The surface of a paraboloid is then calculated:
sfint(1,[u*cos(v),u*sin(v),u^2],[0,1],[0,2*π]) which yields 1/6*(√5*5*π-π)
Arno