Help for a "Surface and Flux integrals" program

11032017, 05:53 PM
(This post was last modified: 11042017 10:05 AM by salvomic.)
Post: #1




Help for a "Surface and Flux integrals" program
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!)... ∫aL√0mic (IT9CLU), HP Prime 50g 41CX 71b 42s 12C 15C  DM42 WP34s :: Prime Soft. Lib 

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