Surface and flux integrals
11-04-2017, 12:17 PM (This post was last modified: 11-06-2017 06:34 PM by Arno K.)
Post: #1
 Arno K Senior Member Posts: 442 Joined: Mar 2015
Surface and flux integrals
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:
#cassfint(f,g,uvals,vvals):=BEGINlocal 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 fgg:=transpose(grad(g,[u,v]));//compute the jacobiangg:=cross(col(gg,1),col(gg,2));IF type(f)==DOM_LIST THEN// in case a vector-function is enteredf:=DOT(f,gg);ELSEf:=f*ABS(gg);//f is a scalarEND;f:=int(f,u,uvals[1],uvals[2])  return int(f,v,vvals[1],vvals[2]);//return the double integralEND;#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
11-04-2017, 01:07 PM
Post: #2
 salvomic Senior Member Posts: 1,392 Joined: Jan 2015
RE: Surface and flux integrals
Thank you, Arno!

Salvo M.

∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
11-04-2017, 01:34 PM
Post: #3
 Arno K Senior Member Posts: 442 Joined: Mar 2015
RE: Surface and flux integrals
(11-04-2017 01:07 PM)salvomic Wrote:  Thank you, Arno!

Salvo M.

Was fun to try mixing cas cammands.
Arno
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