ÅðéöáíåéáêÜ ïëïêëçñþìáôá

1   ÅéóáãùãéêÜ óôïé÷åßá ãéá ôá
        ÅðéöáíåéáêÜ ïëïêëçñþìáôá

Äéáéóèçôéêüò ïñéóìüò åìâáäïý åðéöáíåßáò.

Èá îåêéíÞóïõìå ôçí ìåëÝôç ôùí åðéöáíåéáêþí ïëïêëçñùìÜôùí èåùñþíôáò ìéá åðéöÜíåéá êáé ðñïóðáèþíôáò íá âñïýìå ôï åìâáäüí ôçò. Èåùñïýìå Ýíá ðáñáâïëïåéäÝò ðïõ ðåñéãñÜöåôáé áðü ôçí åîßóùóç 
                g[x, y] = 4 - (x^2) - (y^2).  
×ñçóéìïðïéïýìå êõëéíäñéêÝò óõíôåôáãìÝíåò ãéá ôçí ãñáöéêÞ ôïõ ðáñÜóôáóç ( áíôß ôïõ r êáé è ÷ñçóéìïðïéïýìå ôá óýìâïëá u êáé v).

Clear[P, u, v, surface, g, x, y];
plotpoints = 25;
g[x_, y_] = 4 - (x^2) - (y^2);
{x[u_, v_], y[u_, v_]} = {u Cos[v], u Sin[v]};
P[u_, v_] = {x[u, v], y[u, v], g[x[u, v], y[u, v]]};
surface = 
        ParametricPlot3D[P[u, v], {u, 0, 1}, {v, 0, 2 π}, 
          PlotPoints -> plotpoints];

ÅîåôÜæïíôáò ðñïóåêôéêÜ ôçí ãñáöéêÞ ðáñÜóôáóç ðïõ Ýöôéáîå ç åöáñìïãÞ ìáò ðáñáôçñïýìå üôé ç åðéöÜíåéá åßíáé "öôéáãìÝíç" áðü ðïëëÜ ìéêñÜ êïììÜôéá, êÜèå Ýíá áðü ôá ïðïßá ìïéÜæåé íá åßíáé Ýíá ðáñáëëçëüãñáììï. Åáí õðïëïãßóïõìå ôï åìâáäüí êÜèå êïììáôéïý ôüôå ìðïñïýìå íá ðñïóåããßóïõìå ôï åìâáäüí ôçò åðéöÜíåéáò ðñïóèÝôïíôáò üëá ôá åðéìÝñïõò åìâáäÜ. 

ÊÜíïõìå åê íÝïõ ôçí ãñáöéêÞ ðáñÜóôáóç ôçò åðéöÜíåéáò áõîÜíïíôáò üìùò ôïí áñéèìü ôùí plotpoints (äçëáäÞ, ôï áñéèìü ôùí óçìåßùí ðïõ åðéëÝãïõìå óôï ðåäßï ïñéóìïý ôçò ðáñáìåôñéêïðïßçóçò ãéá íá ó÷çìáôßóïõìå ôçí åðéöÜíåéá).

Clear[P, u, v, surface, g, x, y];
plotpoints = 45;
g[x_, y_] = 4 - (x^2) - (y^2);
{x[u_, v_], y[u_, v_]} = {u Cos[v], u Sin[v]};
P[u_, v_] = {x[u, v], y[u, v], g[x[u, v], y[u, v]]};
surface = 
        ParametricPlot3D[P[u, v], {u, 0, 1}, {v, 0, 2 π}, 
          PlotPoints -> plotpoints];

Tá êïììÜôéá (ðïëý ìéêñÜ ðáñáëëçëüãñáììá) ãßíïíôáé ìéêñüôåñá óå åìâáäüí êáé ðñïóåããßæïõí ðïëý ðåñéóóüôåñï ôçí åðéöÜíåéá.  
Óõíåðþò, åáí õðïëïãßóïõìå ôï åìâáäüí êÜèå åíüò áðü áõôÜ ôá ðáñáëëçëüãñáììá êáé ôá áèñïßóïõìå èá ðÜñïõìå ìéá êáëýôåñç ðñïóÝããéóç ôïõ åìâáäïý ôçò åðéöÜíåéáò. 
Âåâáßùò, ãéá íá âñïýìå áêñéâþò ôï åìâáäüí ôçò åðéöÜíåéáò èá ðñÝðåé íá èåùñÞóïõìå áðåéñïóôÜ ðáñáëëçëüãñáììá êáé íá áèñïßóïõìå ôá åìâáäÜ ôïõò. 
Áõôü (ðñÝðåé íá óáò) èõìßæåé áèñïßóìáôá Riemann êáé ïëïêëÞñùóç êáôÜ Riemann.  

Óôéò åðüìåíåò ãñáöéêÝò ðáñáóôÜóåéò èá ðáßñíïõìå plotpoints=10 áðëÜ êáé ìüíï ãéá íá åßíáé åõêñéíÝò ôï ó÷Þìá ìáò.

Clear[P, u, v, surface, g, x, y];
plotpoints = 10;
g[x_, y_] = 4 - (x^2) - (y^2);
{x[u_, v_], y[u_, v_]} = {u Cos[v], u Sin[v]};
P[u_, v_] = {x[u, v], y[u, v], g[x[u, v], y[u, v]]};
surface = 
        ParametricPlot3D[P[u, v], {u, 0, 1}, {v, 0, 2 π}, 
          PlotPoints -> plotpoints];

Ðåñéïñéæüìáóôå óôç óõíÝ÷åéá óå Ýíá óõãêåêñéììÝíï ðáñáëëçëüãñáììï êáé ðñïóðáèïýìå íá õðïëïãßóïõìå ôï åìâáäüí ôïõ. Ïé ãñáììÝò åðß ôçò åðéöáíåßáò áíáðáñéóôïýí êáìðýëåò üðïõ ìßá áðü ôéò äýï ìåôáâëçôÝò (u Þ v) åßíáé óôáèåñÞ. Ïé êýêëïé óôáèåñïý "ãåùãñáöéêïý ðëÜôïõò" åßíáé êáìðýëåò üðïõ ôï u (äçë. ôï r)  åßíáé óôáèåñü êáé ïé ãñáììÝò óôáèåñïý "ãåùãñáöéêïý ìÞêïõò" åßíáé êáìðýëåò üðïõ ôï v (äçë. ôï è) Ý÷åé êñáôçèåß óôáèåñü. ¸íá "ìéêñü" ðáñáëëçëüãñáììï ó÷çìáôßæåôáé áðü ôçí ôïìÞ 4 ôÝôïéùí êáìðõëþí. 

Õðåíèõìßæïõìå üôé ôï åìâáäüí åíüò ðáñáëëçëïãñÜììïõ äßíåôáé áðü ôçí íüñìá ôïõ åîùôåñéêïý ãéíïìÝíïõ ôùí áíõóìÜôùí ðïõ ôï ó÷çìáôßæïõí, 
          Åìâáäüí ÐáñáëëçëïãñÜììïõ = || X × Y || 
Óõíåðþò åáí ìðïñïýóáìå íá âñïýìå äýï áíýóìáôá ðïõ íá ó÷çìáôßæïõí ôéò ðëåõñÝò ôïõ ðáñáëëçëïãñÜììïõ, ôüôå èá ìðïñïýóáìå íá õðïëïãßóïõìå ôï åìâáäüí ôïõ ðáñáëëçëïãñÜììïõ êáé, ùò åê ôïýôïõ, èá åß÷áìå Ýíá ôñüðï õðïëïãéóìïý ôïõ óõíïëéêïý åìâáäïý ôçò åðéöÜíåéáò. 

Ãéá íá âñïýìå äýï ôÝôïéá áíýóìáôá èá ÷ñçóéìïðïßçóïõìå ôï ãåãïíüò üôé ôï  ðáñáëëçëüãñáììï ó÷çìáôßæåôáé áðü êáìðýëåò ðÜíù óôçí åðéöÜíåéá üðïõ åßôå ôï u  åßôå ôï v åßíáé óôáèåñÜ (äçë. ïé êáìðýëåò åßíáé óõíáñôÞóåéò ìéÜò ìåôáâëçôÞò). Óõíåðþò, åáí âñïýìå ôá åöáðôüìåíá áíýóìáôá êáôÜ ìÞêïò áõôþí ôùí êáìðõëþí óå êÜðïéï óõãêåêñéììÝíï óçìåßï, ôá åöáðôüìåíá áõôÜ áíýóìáôá èá ðñïóåããßæïõí ôï ìÞêïò ôùí ðëåõñþí ôïõ ðáñáëëçëçëïãñÜììïõ. 
Äåßôå ôçí ðáñáêÜôù ãñáöéêÞ ðáñÜóôáóç:

Clear[tanu, tanv, tanvectors, du, dv]
du = (1/plotpoints);
\(dv = \(2\ π\)\/plotpoints; \  (*du\ êáé\ dv\ åßíáé\ óõíôåëåóôÝò\ óêïðüò\ ôùí\ ïðïßùí\ åßíáé\ íá\ êÜíïõí\ ôá\ åöáðôüìåíá\ áíýóìáôá\ íá\ ðñïóåããßæïõí\ ôçò\ ðëåõñÝò\ ôïõ\ ðáñáëëçëïãñÜììïõ*) 

    tanu[u_, v_] = Simplify[du\ ∂\_u P[u, v]];\)
tanv[u_, v_] = Simplify[dv (∂_v) P[u, v]];
tanvectors[u_, v_] := {Arrow[tanu[u, v], Tail -> P[u, v], Red], 
          Arrow[tanv[u, v], Tail -> P[u, v], Blue]};
Show[surface, tanvectors[ .55, 0], 
        ViewPoint -> {3.164, 1.200, 0.031}];

ÂëÝðïõìå ðùò ôá åöáðôüìåíá áíýóìáôá ðñïóåããßæïõí ôéò ðëåõñÝò ôïõ ðáñáëëçëïãñÜììïõ. Ó÷çìáôßæïõìå ôï åîùôåñéêü ãéíüìåíï ôùí äýï áíõóìÜôùí :

Clear[normal, normalvector]
scale = 5;
normal[u_, v_] = 
        Simplify[scale tanu[u, v]\[Cross]tanv[u, v]];
normalvector[u_, 
          v_] := {Arrow[normal[u, v], Tail -> P[u, v], 
            Goldenrod]};
Clear[u0, v0]
{u0, v0} = { .55, 0};
Show[surface, tanvectors[u0, v0], normalvector[u0, v0], 
        ViewPoint -> {3.164, 1.200, 0.031}];

Ôï åìâáäüí ôïõ ðáñáëëçëïãñÜììïõ ðïõ êáèïñßæåôáé áðü ôï ìðëÝ êáé ôï êüêêéíï Üíõóìá åßíáé 

  |Êüêêéíï¢íõóìá × ÌðëÝ¢íõóìá| ôï ïðïßï éóïýôáé ìå:

ÅìâáäüíÐáñáëëçëïãñÜììïõ = 
      PowerExpand[Simplify[Norm[tanu[u, v]\[Cross]tanv[u, v]]]]

ÕðÜñ÷åé ðïëý áðëïýóôåñïò ôñüðïò ãéÜ íá öôÜóïõìå óôï ðáñáðÜíù áðïôÝëåóìá :

Simplify[
      Norm[(∂_u) P[u, v]\[Cross](∂_v) P[u, v]]]

Ç ðáñáðÜíù ðïóüôçôá åßíáé áðëÜ ç íüñìá ôïõ åîùôåñéêïý ãéíïìÝíïõ ôùí ìåñéêþí ðáñáãþãùí ùò ðñïò u  êáé v.  ÁõôÞ ôçí ðïóüôçôá ôçí ïíïìÜæïõìå 
                    ÓõíôåëåóôÞ Åðéöáíåßáò
                    
 Ôï äçëþíïõìå áõôü óôçí åöáñìïãÞ ìáò :

ÓõíôåëåóôÞòÅðéöáíåßáò = 
      Simplify[Norm[(∂_u) P[u, v]\[Cross](∂_v) P[u, 
                v]]]

Åáí óõíõðïëïãßóïõìå êáé ôïõò óõíôåëåóôÝò du dv ðïõ ÷ñçóéìïðïéÞóáìå ãéá íá êÜíïõìå ôá åöáðôüìåíá áíýóìáôá íá ðñïóåããßóïõí ôçò ðëåõñÝò ôïõ ðáñáëëçëïãñÜììïõ Ý÷ïõìå üôé ôï åìâáäüí ôïõ ðáñáëëçëïãñÜììïõ åßíáé áêñéâþò ôï ßäéï ìå 
                 ÓõíôåëåóôÞÅðéöáíåßáò du dv

{ÓõíôåëåóôÞòÅðéöáíåßáò du dv, ÅìâáäüíÐáñáëëçëïãñÜììïõ}

H ðáñáðÜíù ðïóüôçôá äßíåé ôï åìâáäüí ôïõ ìéêñïý êïììáôéïý (ðáñáëëçëïãñÜììïõ). Åßíáé ëïéðüí áíáìåíüìåíï íá âñïýìå ôï åìâáäüí ôçò åðéöÜíåéáò áèñïßæïíôáò ôá åìâáäÜ "áðåéñïóôþí" ðáñáëëçëëïãñÜììùí, äçëáäÞ,

(∫_0^1)(∫_0^(2 π))ÓõíôåëåóôÞòÅðéöáíåßáò \[DifferentialD]v \[DifferentialD]u

Ï ÓõíôåëåóôÞòÅðéöáíåßáò ùò óõíÜñôçóç ôùí u êáé v ìéáò åðéöÜíåéáò ìå ðáñáìÝôñéóç P(u, v) ùò ðñïò ìåôáâëçôÝò u êáé v õðïëïãßæåôáé áðü ôï Mathematica áðü ìßá êáôÜëëçëá ïñéóìÝíç óõíÜñôçóç :  SA[P[u,v],{u,v}] 

Óôçí ðåñßðôùóÞ ìáò (äçëáäÞ ôçò åðéöÜíåéáò ôïõ ðáñáâïëïåéäïýò) Ý÷ïõìå

SA[P[u, v], {u, v}]

SA[P[u, v], {u, v}]  //  Simplify

Ìßá Üëëç êáôÜëëçëá ïñéóìÝíç óõíÜñôçóç NV[P[u,v],{u,v}]  õðïëïãßæåé ôï åãêÜñóéï Üíõóìá óôçí åðéöÜíåéá :

NV[P[u, v], {u, v}]

¼ðùò åßäáìå ðáñáðÜíù:

SA[P[u, v], {u, v}] == Norm[NV[P[u, v], {u, v}]]

ÃåíéêÜ:
ÅÜí  S  åßíáé ìßá åðéöÜíåéá êáé P [u,v]  ìéá ðáñáìÝôñéóÞ ôçò ùò ðñïò ìåôáâëçôÝò  u  êáé v ôüôå ï ÓõíôåëåóôÞòÅðéöáíåßáò ôçò S åßíáé ï áñéèìüò 
                        || ∂(P/∂)u ×  ∂(P/∂)v ||
üðïõ ∂(P/∂)u åßíáé ç ìåñéêÞ ðáñÜãùãïò ôçò P [u,v]  ùò ðñïò u êáé ∂(P/∂)v åßíáé ç ìåñéêÞ ðáñÜãùãïò ôçò P [u,v]  ùò ðñïò u.     

Ãéá ìßá ôõ÷ïýóá åðéöÜíåéá S o ÓõíôåëåóôÞòÅðéöáíåßáò
åßíáé :

Clear[x, y, z, u, v]
SA[{x[u, v], y[u, v], z[u, v]}, {u, v}]

Åßíáé åìöáíÝò üôé ìßá ôÝôïéá ó÷Ýóç äåí áðïìíçìïíåýåôáé. Ïöåßëïõìå íá ãíùñßæïõìå üìùò ðùò ðáñÜãåôáé.

ÅãêÜñóéï Üíõóìá êáé ðñïóáíáôïëéóìüò
 åðéöÜíåéáò.

Óôçí ðáñáðÜíù äéáäéêáóßá åýñåóçò ÷ñçóéìïðïéÞóáìå ôçí íüñìá ôïõ åãêÜñóéïõ óôçí åðéöÜíåéá P(u,v) áíýóìáôïò ôï ïðïßï, üðùò åßäáìå, åßíáé ßóï ìå  
                         || ∂(P/∂)u ×  ∂(P/∂)v ||
¼ìùò, åãêÜñóéï óôçí åðéöÜíåéá P(u,v) åßíáé êáé ôï Üíõóìá
                            ∂(P/∂)v ×  ∂(P/∂)u 
êáé éó÷ýåé 
                        ∂(P/∂)v ×  ∂(P/∂)u = - ∂(P/∂)u ×  ∂(P/∂)v
Âåâáßùò, üôáí èåùñïýìå íüñìåò ôüôå äåí Ý÷åé óçìáóßá ç óåéñÜ ôùí ìåôáâëçôþí Þ, éóïäýíáìá, ç ðñïåðéëïãÞ åíüò 
áðü ôá äýï åãêÜñóéá äéáíýóìáôá  ∂(P/∂)v ×  ∂(P/∂)u,  ∂(P/∂)u ×  ∂(P/∂)v.
¼ìùò, óôçí óõíÝ÷åéá èá èåùñÞóïõìå ïëïêëçñþìáôá óôá ïðïßá èá åìðëÝêåôáé ôï åãêÜñóéï Üíõóìá êáé ü÷é ç íüñìá ôïõ. Ðéï óõãêåêñéììÝíá, èá ÷ñçóéìïðïéÞóïõìå ôï 
ìïíáäéáßï åãêÜñóéï Üíõóìá
                             n = ((  ∂(P/∂)u ×  ∂(P/∂)v )/( || ∂(P/∂)u ×  ∂(P/∂)v ||))
Èá ðñÝðåé ëïéðüí íá åðéëÝîïõìå ìßá óåéñÜ ãéá ôéò ìåôáâëçôÝò u,v  ç ïðïßá êáé èá êáèïñßæåé ôï "ðñïò ôá Ýîù" åãêÜñóéï Üíõóìá n óôçí åðéöÜíåéá  P(u,v). Ìéá åðéöÜíåéá ìáæß ìå ìéá ôÝôïéá åðéëïãÞ èá ëÝãåôáé ðñïóáíáôïëéóìÝíç åðéöÜíåéá.

Ðñïóï÷Þ: Ìéá ôÝôïéá åðéëïãÞ äåí åßíáé ðÜíôá åöéêôÞ. Äåßôå ôçí ðáñáêÜôù åðéöÜíåéá.

Óôï åîÞò èá ðåñéïñéóôïýìå óå åðéöÜíåéåò ðïõ åßíáé ðñïóáíáôïëßóéìåò. Ìå Üëëá ëüãéá, óôç ëÝîç åðéöÜíåéá èá åìðåñéÝ÷åôáé êáé ï ðñïóáíáôïëéóìüò ôçò, äçëáäÞ, ìéá åðéëïãÞ åãêáñóßïõ áíýóìáôïò. Äåßôå ôçí ðáñáêÜôù åðéöÜíåéá.

Clear[normal, normalvector]
scale = 7;
scaleminus = -7;
normal[u_, v_] = 
        Simplify[
          scale tanu[u, v]\[Cross]tanv[u, v]];
normalminus[u_, v_] = 
        Simplify[scaleminus tanu[u, v]\[Cross]tanv[u, v]];
normalvector[u_, 
          v_] := {Arrow[normal[u, v], Tail -> P[u, v], 
            Red]};
normalvectorminus[u_, 
          v_] := {Arrow[normalminus[u, v], Tail -> P[u, v], 
            Blue]};
Clear[u0, v0]
{u0, v0} = { .55, 0};
Show[surface, normalvector[u0, v0], normalvectorminus[u0, v0], 
        ViewPoint -> {3.164, 0.500, -3.031}];

Ï ÓõíôåëåóôÞòÅðéöáíåßáò ãéá ìéá åðéöÜíåéá ðïõ ïñßæåôáé áðü åîßóùóç ôçò ìïñöÞò 
                             z=f[x,y].

Ç ðáñáìåôñéêïðïßçóç ôçò åðéöÜíåéáò z=f (x,y)  äßíåôáé áðü ôçí åîßóùóç 
                     P(x,y) = (x, y, f (x,y))
Âñßóêïõìå ôïí
                     ÓõíôåëåóôÞÅðéöáíåßáò
ï ïðïßïò, õðåíèõìßæïõìå,  åßíáé ßóïò ìå 

                         || ∂(P/∂)x ×  ∂(P/∂)y ||

Clear[x, y, f]
S[x_, y_] = SA[{x, y, f[x, y]}, {x, y}]

ÅîÞãçóç: Ôï Mathematica Ýêáíå, óôï ðáñáðÜíù cell, ôïí åîÞò (óõìâïëéêü) õðïëïãéóìü: 
       ∂(P/∂)x= (1, 0,∂(f/∂)x)  êáé  ∂(P/∂)y= (0, 1,∂(f/∂)y)
ïðüôå       
            || ∂(P/∂)x ×  ∂(P/∂)y || = || ({{({{i, j, k}, {1, 0, ∂(f/∂)x}, {0, 1, ∂(f/∂)y}})}, { }}) || = || (-∂(f/∂)x, -∂(f/∂)y, 1 ||
                         = \(TraditionalForm\`\@\(\(1\)\(+\)\(\((∂f\/∂x)\)\^2\)\(+\)\(\((∂f\/∂y)\)\^2\)\(\ \)\)\)

2    Ïñéóìïß 
         Åðéöáíåéáêþí ïëïêëçñùìÜôùí

Ïñéóìüò åìâáäïý åðéöáíåßáò ùò Åðéöáíåéáêü ïëïêëÞñùìá

¸óôù  S   ìßá åðéöÜíåéá óôïí R\(TraditionalForm\`\^3\) êáé  P : R-> R\(TraditionalForm\`\^3\)  ìéá ðáñáìÝôñéóÞ ôçò ùò ðñïò ìåôáâëçôÝò  u  êáé v ðïõ áíÞêïõí óôï õðïóýíïëï R ôïõ R\(TraditionalForm\`\^2\), äçëáäÞ,
              S = { P(u,v) : (u,v) ∈ R}
 ¼ðùò åßäáìå ðáñáðÜíù, ï ÓõíôåëåóôÞòÅðéöáíåßáò ôçò S åßíáé ï áñéèìüò 
                        || ∂(P/∂)u ×  ∂(P/∂)v ||
üðïõ ∂(P/∂)u åßíáé ç ìåñéêÞ ðáñÜãùãïò ôçò P (u,v)  ùò ðñïò u êáé ∂(P/∂)v åßíáé ç ìåñéêÞ ðáñÜãùãïò ôçò P (u,v)  ùò ðñïò v êáé

ôï Åìâáäüí ôçò S  åßíáé :     

                   ∫(∫_R)|| ∂(P/∂)u ×  ∂(P/∂)v || du dv

Ïñéóìüò Åðéöáíåéáêïý ïëïêëçñþìáôïò óõíÜñôçóçò (âáèìùôïý ðåäßïõ)

¸óôù  S   ìßá åðéöÜíåéá óôïí R\(TraditionalForm\`\^3\) êáé  P : R-> R\(TraditionalForm\`\^3\)  ìéá ðáñáìÝôñéóÞ ôçò ùò ðñïò ìåôáâëçôÝò  u  êáé v, äçëáäÞ,
              S = { P(u,v) : (u,v) ∈ R}
 Ùò ãíùóôüí, ï ÓõíôåëåóôÞòÅðéöáíåßáò ôçò S åßíáé ï áñéèìüò 
                        || ∂(P/∂)u ×  ∂(P/∂)v ||
üðïõ ∂(P/∂)u åßíáé ç ìåñéêÞ ðáñÜãùãïò ôçò P (u,v)  ùò ðñïò u êáé ∂(P/∂)v åßíáé ç ìåñéêÞ ðáñÜãùãïò ôçò P (u,v) ùò ðñïò v.

Åáí ø : S -> R åßíáé ìßá äéáöïñßóéìç óõíÜñôçóç ôüôå ôï Åðéöáíåéáêü ïëïêëÞñùìá ôçò ø åðß ôçò S  åßíáé :     

              (∫_S) ø  =∫(∫_R) ø (P (u, v)) || ∂(P/∂)u ×  ∂(P/∂)v || du dv

Ïñéóìüò Åðéöáíåéáêïý ïëïêëçñþìáôïò äéáíõóìáôéêïý ðåäßïõ.

¸óôù  S   ìßá åðéöÜíåéá óôïí R\(TraditionalForm\`\^3\) êáé  P : R-> R\(TraditionalForm\`\^3\)  ìéá ðáñáìÝôñéóÞ ôçò ùò ðñïò ìåôáâëçôÝò  u  êáé v, äçëáäÞ,
              S = { P(u,v) : (u,v) ∈ R}
¸óôù U  áíïéêôü óýíïëï óôïí R\(TraditionalForm\`\^3\) ðïõ ðåñéÝ÷åé ôçí S, êáé Ýóôù F : U -> R\(TraditionalForm\`\^3\) äéáöïñßóéìï äéáíõóìáôéêü ðåäßï. Ôüôå, ôï åóùôåñéêü ãéíüìåíï
                F . n :=F(P (u, v)) .n(P (u, v)) 
åßíáé ìßá äéáöïñßóéìç óõíÜñôçóç åðß ôçò S  (ç ïðïßá ìáò äßíåé ôçí ôéìÞ ôçò ðñïâïëÞò ôïõ F  óôçí äéåýèõíóç ôïõ åãêÜñóéïõ áíýóìáôïò n.

Ôï Åðéöáíåéáêü ïëïêëÞñùìá ôïõ F  åðß ôçò S  åßíáé :     
              (∫_S)  F :=∫(∫_R) F(P (u, v)) .n(P (u, v)) || ∂(P/∂)u ×  ∂(P/∂)v || du dv
              
ÅðåéäÞ ãéá ôï åãêÜñóéï Üíõóìá n éó÷ýåé
                          n = ((  ∂(P/∂)u ×  ∂(P/∂)v )/( || ∂(P/∂)u ×  ∂(P/∂)v ||))
ï ïñéóìüò ôïõ åðéöáíåéáêïý ïëïêëçñþìáôïò ôïõ F  åðß ôçò S  ãñÜöåôáé 

          (∫_S)  F :=∫(∫_R) F(P (u, v)) . (∂(P/∂)u ×  ∂(P/∂)v) dudv

3    Õðïëïãéóìïß  åðéöáíåéáêþí 
         ïëïêëçñùìÜôùí

ÐáñÜäåéãìá 1:  Åýñåóç åìâáäïý óöáßñáò áêôßíáò  R

Áöïý ç åðéöÜíåéÜ ìáò åßíáé óöáßñá ÷ñçóéìïðïéïýìå óöáéñéêÝò óõíôåôáãìÝíåò (ñ, ϕ, θ).  ÈÝôïõìå ôçí óõíôåôáãìÝíç ñ ßóç ìå ôçí óôáèåñÜ R áöïý ìáò åíäéáöÝñåé ç åðéöÜíåéá ôçò óöáßñáò êáé ü÷é ôï åóùôåñéêü ôçò.

Clear[x, y, z, ñ, ϕ, θ, S, R]
ñ = R;
x[ϕ_, θ_] = ñ Sin[ϕ] Cos[θ];
y[ϕ_, θ_] = ñ Sin[ϕ] Sin[θ];
z[ϕ_, θ_] = ñ Cos[ϕ];
S[ϕ_, θ_] = 
      PowerExpand[
        Simplify[
          SA[{x[ϕ, θ], y[ϕ, θ], 
              z[ϕ, θ]}, {ϕ, θ}]]]

Ï ÓõíôåëåóôÞòÅðéöáíåßáò åßíáé (R^2) Sin[ϕ].  
Óõíåðþò, ç åðéöÜíåéá ôçò óöáßñáò åßíáé :

ÅðéöÜíåéáÓöáßñáò = (∫_0^π)(∫_0^(2 π))(R^2) Sin[ϕ] \[DifferentialD]θ \[DifferentialD]ϕ

ÐáñÜäåéãìá 2:  Åýñåóç åìâáäïý ôïõ ôìÞìáôïò ôïõ ðáñáâïëïåéäïýò z = (x^2)+(y^2) ðïõ öñÜóóåôáé áðü ôï åðßðåäï z = 9

Ôï ðñþôï ðñÜãìá ðïõ ðñÝðåé íá êÜíïõìå åßíáé íá ðáñáìåôñéêïðïéÞóïõìå ôçí åðéöÜíåéÜ ìáò.  
×ñçóéìïðïéïýìå ðñþôá êáñôåóéáíÝò óõíôåôáãìÝíåò.

Õðåíèýìéóç: Ç ðáñáìåôñéêïðïßçóç ùò ðñïò x êáé y ìéáò ôõ÷áßáò åðéöÜíåéáò ôçò ìïñöÞò z=f(x,y) åßíáé  
                     P(x,y) = (x, y, f (x,y)). 
 
Ôá üñéá ïëïêëÞñùóçò ãéá ôá x êáé y åßíáé: 
        áöïý 0 ≤ z ≤ 9, x êáé y ðñÝðåé íá âñßóêïíôáé ìÝóá 
        óôï ÷ùñßï (x^2)+(y^2)≤ 9, ôï ïðïßï åßíáé äßóêïò áêôßíáò 3.  Óôçí óõíÝ÷åéá ðñÝðåé íá âñïýìå ôïí      
                        ÓõíôåëåóôÞÅðéöáíåßáò

Clear[f, x, y, S, f]
f[x_, y_] = (x^2) + (y^2);
S[x_, y_] = SA[{x, y, f[x, y]}, {x, y}]

Óôçí óõíÝ÷åéá ïëïêëçñþíïõìå åðß ôïõ ÷ùñßïõ (x^2)+(y^2)≤ 9.

NIntegrate[
      S[x, y], {x, -3, 
        3}, {y, -(Sqrt[9 - (x^2)]), (Sqrt[9 - (x^2)])}]

ÄåäïìÝíïõ üôé ïëïêëçñþíïõìå åðß åíüò äßóêïõ áêôßíáò 3, åßíáé ëïãéêü íá óêåöèïýìå üôé èåùñþíôáò ðïëéêÝò óõíôåôáãìÝíåò ï õðïëïãéóìüò åßíáé áðëïýóôåñïò. ÊÜíïõìå ëïéðüí ôïí ßäéï õðïëïãéóìü ìå ðïëéêÝò óõíôåôáãìÝíåò.

ÊáôÜ ôçí ìåôáôñïðÞ ôïõ ðáñáðÜíù ïëïêëçñþìáôïò óå ðïëéêÝò óõíôåôáãìÝíåò ðñÝðåé íá åéóÜãïõìå êáé ôçí ïñßæïõóá ôçò ÉáêùâéáíÞò ôïõ ìåôáó÷çìáôéóìïý áðü êáñôåóéáíÝò óå ðïëéêÝò.

Clear[x, y, r, t]
x[r_, t_] = r Cos[t];
y[r_, t_] = r Sin[t];
J[r_, t_] = jacobiandet[{x[r, t], y[r, t]}, {r, t}]

NIntegrate[
      J[r, t] S[x[r, t], y[r, t]], {t, 0, 2  π}, {r, 0, 3}]

Èá ìðïñïýóáìå âåâáßùò íá ÷ñçóéìïðïéÞóïõìå êõëéíäñéêÝò óõíôåôáãìÝíåò åî' áñ÷Þò:

Clear[f, x, y, S, r, t, J]
z[x_, y_] = (x^2) + (y^2);
x[r_, t_] = r Cos[t];
y[r_, t_] = r Sin[t];
z[r_, t_] = z[x[r, t],  y[r, t]]
S[r_, t_] = 
      SA[{x[r, t], y[r, t], z[r, t]}, {r, t}] // Simplify

Ðáñáôçñïýìå üôé ç ïñßæïõóá ôçò ÉáêùâéáíÞò ôïõ ìåôáó÷çìáôéóìïý áðü êáñôåóéáíÝò óå ðïëéêÝò åìðåñéÝ÷åôáé óôïí  ÓõíôåëåóôÞÅðéöáíåßáò üôáí áõôüò õðïëïãéóôåß ìå ôçí ðáñáìåôñéêïðïßçóç ôùí êõëéíäñéêþí óõíôåôáãìÝíùí, äçëáäÞ, 
             || ∂(P/∂)r ×  ((∂P)/(∂è)) || = \(TraditionalForm\`\@\(r\^2 + 4  r\^2\) = \ \(\@\(1 + 4 \((x\^2 + y\^2)\)\)\ r = r \@\( 1 + 4  r\)\)\)
üðïõ P(r, è) =  (r Cos[t], r Sin[t], r\(TraditionalForm\`\^2\)).

(∫_0^3)(∫_0^(2 π)) 
          S[r, t] \[DifferentialD]t \[DifferentialD]r // N

ÐáñÜäåéãìá 3: Õðïëïãéóìüò åðéöáíåéáêïý ïëïêëçñþìáôïò 
          ∫(∫_S)\(TraditionalForm\`\(\_\ \) \((x\^2\)\) + (y^2))  
åðß ðáñáìåôñéêïðïéçìÝíçò åðéöÜíåéáò S

¸óôù ç åðéöÜíåéá S ðïõ äßíåôáé áðü ôçí ðáñáìåôñéêÞ åîßóùóç 
                     P(u,v) = (u Cos[v], -u Sin[v], v)
                1 ≤ u ≤ 3 ,   0 ≤ v ≤ ((3  π)/2)

Clear[x, y, z, u, v]
x[u_, v_] = u Cos[v];
y[u_, v_] = -u Sin[v];
z[u_, v_] = v;
{{ulow, uhigh}, {vlow, vhigh}} = {{1, 
            3}, {0, ((3 π)/2)}};
surface = 
        ParametricPlot3D[{x[u, v], y[u, v], z[u, v]}, {u, ulow, uhigh}, {v, 
            vlow, vhigh}, ViewPoint -> CMView, Boxed -> False, 
          AxesLabel -> {"x", "y", "z"}];

Ôï ïëïêëÞñùìá ∫(∫_S) (x^2)+(y^2) äßíåôáé áðü ôïí ôýðï 

∫(∫_S) (x^2)+(y^2)= ∫(∫_R) (x[u, v]^2 + y[u, v]^2)||∂(P/∂)u×∂(P/∂)v||du dv

üðïõ R åßíáé ôï ÷ùñßï [1,3]×[0,((3  π)/2)].

Õðïëïãßæïõìå ôïí ÓõíôåëåóôÞÅðéöáíåßáò 

                      S[u,v] = ||∂(P/∂)u×∂(P/∂)v||

Clear[S]
S[u_, v_] = 
      Simplify[SA[{x[u, v], y[u, v], z[u, v]}, {u, v}]]

Óõíåðþò, ôï æçôïýìåíï ïëïêëÞñùìá åßíáé 

∫(∫_S) (x^2)+(y^2)=
∫(∫_R) (u^2) (Sqrt[1 + 
                  (u^2)]) du dv = (∫_0^(((3  π)/2)))(∫_1^3) 
                (u^2) (Sqrt[1 + 
                  (u^2)]) du dv

Ìå ôï Mathematica âñßóêïõìå åýêïëá ôçí áðÜíôçóç

integral = (∫_vlow^vhigh)(∫_ulow^uhigh)(x[u, v]^2 +  y[u, v]^2) S[u, v] \[DifferentialD]u \[DifferentialD]v // 
        N

ÐáñÜäåéãìá 4: Åýñåóç åìâáäïý ôïõ ôìÞìáôïò ôïõ åðéðÝäïõ x+ 2y + z=4 ðïõ âñßóêåôáé ìÝóá óôïí êýëéíäñï (x^2)+(y^2)=4

ÊÜíïõìå ôïí õðïëïãéóìü ÷ñçóéìïðïéþíôáò ðñþôá êáñôåóéáíÝò óõíôåôáãìÝíåò. Âñßóêïõìå  ðñþôá ôïí ÓõíôåëåóôÞÅðéöáíåßáò 

                       ||∂(P/∂)x×∂(P/∂)y|| 
üðïõ P(x,y) = (x, y, f (x,y)) ìå f (x,y) = 4 - x - 2  y. ¼ðùò Ý÷ïõìå áðïäåßîåé ðáñáðÜíù 
            ||∂(P/∂)x×∂(P/∂)y|| = (Sqrt[1+(((∂(f/∂)x))^2)+ (((∂(f/∂)y))^2)])= (Sqrt[6]).

Clear[f, x, y, S]
f[x_, y_] = 4 - x - 2 y;
\(S[x_, y_] = SA[{x, y, f[x, y]}, {x, y}]
\)

Ãéá íá åßìáóôå ìÝóá óôïí êýëéíäñï áêôßíáò 2 ðñÝðåé íá ïëïêëçñþóïõìå ôá x êáé y åðß ôïõ äßóêïõ áêôßíáò 2 ïðüôå, ôï æçôïýìåíï ïëïêëÞñùìá åßíáé
                    (∫_(-2)^2)(∫_(-(Sqrt[4 - (x^2)]))^((Sqrt[4 - (x^2)]))) (Sqrt[6]) dy dx

Clear[f, x, y, S]
f[x_, y_] = 4 - x - 2 y;
S[x_, y_] = SA[{x, y, f[x, y]}, {x, y}]
(∫_(-2)^2)(∫_(-(Sqrt[4 - (x^2)]))^((Sqrt[4 - (x^2)])))S[x, y] \[DifferentialD]y \[DifferentialD]x

Ôïí ðáñáðÜíù õðïëïãéóìü ìðïñïýìå íá ôïí ðñáãìáôïðïéÞóïõìå ÷ñçóéìïðïéþíôáò êõëéíäñéêÝò óõíôåôáãìÝíåò. Ãéá ôá x êáé y Ý÷ïõìå
            x= r Cos[θ]    êáé       y= r Sin[θ].  
Ãéá ôï z, ôçí ôñßôç óõíôåôáãìÝíç, Ý÷ïõìå  áðü ôçí åîßóùóç ôïõ åðéðÝäïõ 
           z = 4 -  r Cos[θ]  - 2 r Sin[θ].  
Õðïëïãßæïõìå ðñþôá ôïí ÓõíôåëåóôÞÅðéöáíåßáò

\(
Clear[f, x, y, S, r, θ, J]\)
f[x_, y_] = 4 - x - 2  y
x[r_, θ_] = r Cos[θ];
\(\(y[r_, θ_] = r\ Sin[θ];\)
\)
S[r_, θ_] = 
      SA[{x[r, θ], y[r, θ], 
            f[x[r, θ], y[r, θ]]}, {r, θ}] // 
        Simplify

Ïðüôå ôï æçôïýìåíï ïëïêëÞñùìá åßíáé

                    \(∫\_0\%2\(∫\_0\%\(2\ π\)\ \@6\ \(\@r\^2\) dθ dr\)\)

×ñçóéìïðïéþíôáò ôï Mathematica åýêïëá õðïëïãßæïõìå

(∫_0^2)(∫_0^(2 π)) 
        S[r, θ] \[DifferentialD]θ \[DifferentialD]r

ÐáñÜäåéãìá 5: Åýñåóç åìâáäïý ôçò åðéöÜíåéáò ðïõ äßíåôáé áðü ôéò ðáñáìåôñéêÝò åîéóþóåéò :
	x = u v
	y= u + v
	z= u -v         ìå    (u^2)+(v^2) ≤ 1

Ùò óõíÞèùò, âñßóêïõìå ôïí ÓõíôåëåóôÞÅðéöáíåßáò 
                       ||∂(P/∂)u×∂(P/∂)v|| 
üðïõ P  åßíáé ç äïèåßóá ðáñáìåôñéêïðïßçóç    
                 P(u, v) = (u v, u + v, u - v)
ÊÜíïíôáò ðñÜîåéò Ý÷ïõìå 
      ||∂(P/∂)u×∂(P/∂)v|| = |({{i, j, k}, {v, 1, 1}, {u, 1, -1}})|= ... = (Sqrt[4+(((-u + v))^2)+ (((u + v))^2)])

Clear[x, y, z, u, v, S, area]
x = u v;
y = u + v;
z = u - v;
S[u_, v_] = SA[{x, y, z}, {u, v}] // Simplify

Ôï æçôïýìåíï ïëïêëÞñùìá åßíáé

                    (∫_(-1)^1)(∫_(-(Sqrt[1 - (u^2)]))^((Sqrt[1 - (u^2)]))) \(\(\@\(4 + \((\(-u\) + v)\)\^2 + \((u + v)\)\^2\)\) dv du\)

×ñçóéìïðïéþíôáò ôï Mathematica åýêïëá õðïëïãßæïõìå

area = 
      NIntegrate[
        S[u, v], {u, -1, 
          1}, {v, -(Sqrt[1 - (u^2)]), (Sqrt[1 - (u^2)])}]

Óôçí ðáñáêÜôù ãñáöéêÞ ðáñÜóôáóç âëÝðïõìå ôçí åðéöÜíåéá. ×ñçóéìïðïéïýìå (ãéá ôçí ãñáöéêÞ ðáñÜóôáóç) ôçí ðáñáìåôñéêïðïßçóç ðïõ ðñïêýðôåé áíôéêáèéóôþíôáò
                  u  =  r Cos t ,  v = r Sin t
äçëáäÞ,

P(r, t) =  ((r^2) Cos t  Sin t,  
          r Cos t  + r Sin t,  r Cos t  - r Sin t)

üðïõ (r, 
          t) ∈ [0, 1]×[0, 2  π], áöïý (u^2)+(v^2) ≤ 1.
×ñçóéìïðïéïýìå äéáöïñåôéêÜ óçìåßá ðáñáôÞñçóçò (view points) ãéá íá Ý÷ïõìå êáëýôåñç åéêüíá ôçò åðéöÜíåéáò.

ps = 
        ParametricPlot3D[{r Cos[t]*r Sin[t], r Cos[t] + r Sin[t], 
            r Cos[t] - r Sin[t]}, {r, 0, 1}, {t, 0, 
            2  Pi}];
Show[ps, ViewPoint -> {3.164, 1.200, 0.031}]; 
    Show[ps, ViewPoint -> {1.164, -2.200, 0.031}]; 
    Show[ps, ViewPoint -> {4.164, 3.200, 3.031}];

Ìðïñïýìå (öõóéêÜ) íá õðïëïãßóïõìå ôï æçôïýìåíï ïëïêëÞñùìá ÷ñçóéìïðïéþíôáò ôçí ðáñáìåôñéêïðïßçóç ðïõ öôéÜîáìå ãéá ôçí ãñáöéêÞ ðáñÜóôáóç :
 
P(r, t) =  ((r^2) Cos t  Sin t,  
          r Cos t  + r Sin t,  r Cos t  - r Sin t)

ìå (r, 
          t) ∈ [0, 1]×[0, 
              2  π]. 

Ï ÓõíôåëåóôÞòÅðéöáíåßáò  åßíáé ||∂(P/∂)r×∂(P/∂)t||, êáé ôïí õðïëïãßæïõìå ìå ôï Mathematica

Clear[P, r, t, S, area1]

Clear[P, r, t, S, area1]
P[r_, t_] = {r Cos[t]*r Sin[t], r Cos[t] + r Sin[t], 
        r Cos[t] - r Sin[t]}; 
    S[r_, t_] = SA[P[r, t], {r, t}] // Simplify

Tï æçôïýìåíï åìâáäüí åßíáé \(∫\_0\%1\(∫\_0\%\(2\ π\)\ \@2\ \(\@\(\(r\^2\) \((2 + r\^2)\)\)\) dt dr\)\)

area1 = NIntegrate[S[r, t], {r, 0, 1}, {t, 0, 2  Pi}]

ÐáñÜäåéãìá 6: Åýñåóç åðéöáíåéáêïý ïëïêëçñþìáôïò ôïõ Äéáíõóìáôéêïý ðåäßïõ F(x,y,z) = (x,y,0) åðß ôçò óöáßñáò (x^2)+(y^2)+(z^2)=(a^2), a>0

Bñßóêïõìå ðñþôá ôï åãêÜñóéï Üíõóìá óôçí åðéöÜíåéá. Ðñïò ôïýôï, èåùñïýìå ôçí ðáñáìåôñéêïðïßçóç ôçò óöáßñáò ìå óöáéñéêÝò óõíôåôáãìÝíåò, äçëáäÞ, 

P(ϕ, è) = (a Sin[ϕ]  Cos[è], a Sin[ϕ] Sin[è], a Cos[ϕ])

Clear[x, y, z, ñ, ϕ, θ, a]
ñ = a;
x[ϕ_, θ_] = ñ Sin[ϕ] Cos[θ];
y[ϕ_, θ_] = ñ Sin[ϕ] Sin[θ];
z[ϕ_, θ_] = ñ Cos[ϕ];
Simplify[
      NV[{x[ϕ, θ], y[ϕ, θ], 
          z[ϕ, θ]}, {ϕ, θ}]]

Ðáñáôçñïýìå üôé ôï åãêÜñóéï Üíõóìá åßíáé ðïëëáðëÜóéï ôïõ áíýóìáôïò èÝóçò êáé ìÜëéóôá èåôéêü ðïëëáðëÜóéï äéüôé a > 0 êáé \(TraditionalForm\`0 ≤ ϕ ≤ \(\(π\)\(.\)\)\)
Õðïëïãßæïõìå ôþñá ôï åóùôåñéêü ãéíüìåíï ôïõ åãêÜñóéïõ áíýóìáôïò óå Ýíá ôõ÷üí óçìåßï ìå ôçí ôéìÞ (äéÜíõóìá) ôïõ äéáíõóìáôéêïý ìáò ðåäßïõ F.

Clear[F]
F = {x[ϕ, θ], y[ϕ, θ], 
          0};
Simplify[
      F . NV[{x[ϕ, θ], y[ϕ, θ], 
            z[ϕ, θ]}, {ϕ, θ}]]

Ç åöáñìïãÞ Ýêáíå ôïí åîÞò õðïëïãéóìü: 
   (({{a Cos[θ] Sin[ϕ],, \(\(a\ Sin[ϕ]\ Sin[θ]\)\(,\)\), 0}}))·(({{(a^2) Cos[θ] Sin[ϕ]^2}, {(a^2) Sin[θ] Sin[ϕ]^2}, {(a^2) Cos[ϕ] Sin[ϕ]}}))=
     = (a^3) Cos[θ]^2 Sin[ϕ]^3 + (a^3) Sin[θ]^2 Sin[ϕ]^3= (a^3) Sin[ϕ]^3 
Óõíåðþò ôï æçôïýìåíï ïëïêëÞñùìá åßíáé

∫(∫_R) F(P (u, v)) . (∂(P/∂)u ×  ∂(P/∂)v) dudv = 
                        = \(∫\_0\%\(2  π\)\(∫\_0\%π a\^3\ \ \(Sin[ϕ]\^3\) dϕ dθ\)\)

(∫_0^(2  π))(∫_0^π) (a^3)  Sin[ϕ]^3 \[DifferentialD]ϕ \[DifferentialD]θ

4  ÁóêÞóåéò

Íá âñåßôå ôï åìâáäü ôïõ ôìÞìáôïò ôïõ åðéðÝäïõ 2x+ 3y + z=5 ðïõ âñßóêåôáé ìÝóá óôïí êýëéíäñï (x^2)+(y^2)=(a^2).

1) ÐáñáìåôñéêïðïéÞóôå ôçí åðéöÜíåéá  z=g[x,y]= 5-2x-3y êáé âñåßôå ôïí ÓõíôåëåóôÞÅðéöáíåßáò SA[x,y].

2) Õðïëïãßóôå ôï æçôïýìåíï åìâáäüí.

ËÕÓÇ

Âñßóêïõìå  ðñþôá ôïí ÓõíôåëåóôÞÅðéöáíåßáò 

                       ||∂(P/∂)x×∂(P/∂)y|| 
üðïõ P åßíáé ç ðáñáìÝôñéóç  

         P(x,y) = (x, y, f (x,y)) ìå f (x,y) = 5 - 2  x - 3  y. 
         
¼ðùò Ý÷ïõìå áðïäåßîåé ðáñáðÜíù 
            ||∂(P/∂)x×∂(P/∂)y|| = (Sqrt[1+(((∂(f/∂)x))^2)+ (((∂(f/∂)y))^2)])= (Sqrt[14]).

Clear[f, x, y, S]
f[x_, y_] = 5 - 2  x - 3 y;
S[x_, y_] = SA[{x, y, f[x, y]}, {x, y}]

Ãéá íá åßìáóôå ìÝóá óôïí êýëéíäñï áêôßíáò a ðñÝðåé íá ïëïêëçñþóïõìå ôá x êáé y åðß ôïõ äßóêïõ áêôßíáò a ïðüôå, ôï æçôïýìåíï ïëïêëÞñùìá åßíáé
                    (∫_(-a)^a)(∫_(-(Sqrt[(a^2) - (x^2)]))^((Sqrt[(a^2) - (x^2)]))) (Sqrt[14]) dy dx

Clear[f, x, y, S, a]
f[x_, y_] = 5 - 2  x - 3 y;
S[x_, y_] = SA[{x, y, f[x, y]}, {x, y}]
(∫_(-(Sqrt[(a^2) - (x^2)]))^((Sqrt[(a^2) - (x^2)])))S[x, 
          y] \[DifferentialD]y
(∫_(-a)^a)(∫_(-(Sqrt[(a^2) - (x^2)]))^((Sqrt[(a^2) - (x^2)])))S[x, y] \[DifferentialD]y \[DifferentialD]x

Íá âñåßôå ôï åìâáäü ôïõ ôìÞìáôïò ôïõ ðáñáâïëïåéäïýò x = (y^2) + (z^2) ðïõ âñßóêåôáé ìÝóá óôïí êýëéíäñï (y^2)+(z^2)=9.

1) ÐáñáìåôñéêïðïéÞóôå ôçí åðéöÜíåéá x=g[y,z]= (y^2) + (z^2) êáé âñåßôå ôïí ÓõíôåëåóôÞÅðéöáíåßáò SÁ[y,z].

2)  Õðïëïãßóôå ôï åìâáäüí ôïõ ðáñáâïëïåéäïýò.

ËÕÓÇ

Âñßóêïõìå  ðñþôá ôïí ÓõíôåëåóôÞÅðéöáíåßáò 

                       ||∂(P/∂)x×∂(P/∂)y|| 
üðïõ P  åßíáé ç ðáñáìåôñéêïðïßçóç

         P(y,z) = (g (y,z),y, z) ìå g (y,z) = (y^2) + (z^2). 
         
¼ðùò Ý÷ïõìå áðïäåßîåé ðáñáðÜíù 

 ||∂(P/∂)y×∂(P/∂)z|| = (Sqrt[1+(((∂(g/∂)y))^2)+ (((∂(g/∂)z))^2)])= (Sqrt[1 + 4 (y^2) + 4 (z^2)]).

Clear[g, x, y, z, S]
g[y_, z_] = (z^2) + (y^2);
S[y_, z_] = SA[{g[y, z], y, z}, {y, z}]

Ãéá íá åßìáóôå ìÝóá óôïí êýëéíäñï áêôßíáò 3 ðñÝðåé íá ïëïêëçñþóïõìå ôá y êáé z åðß ôïõ äßóêïõ áêôßíáò 3 óôï yz  åðßðåäï ïðüôå, ôï æçôïýìåíï ïëïêëÞñùìá åßíáé
                    (∫_(-3)^3)(∫_(-(Sqrt[(3^2) - (z^2)]))^((Sqrt[(3^2) - (z^2)]))) \(\@\(1 + 4  z\^2 + 4  y\^2\)\ dy dz\)

Clear[g, x, y, z, S]
g[y_, z_] = (z^2) + (y^2);
S[y_, z_] = SA[{g[y, z], y, z}, {y, z}]
(∫_(-(Sqrt[9 - (z^2)]))^((Sqrt[9 - (z^2)])))S[y, 
          z] \[DifferentialD]y
(∫_(-3)^3)(∫_(-(Sqrt[9 - (z^2)]))^((Sqrt[9 - (z^2)])))S[y, z] \[DifferentialD]y \[DifferentialD]z // Simplify

Åíáëëáêôéêüò õðïëïãéóìüò

Ôïí ðáñáðÜíù õðïëïãéóìü ìðïñïýìå íá ôïí ðñáãìáôïðïéÞóïõìå ÷ñçóéìïðïéþíôáò êõëéíäñéêÝò óõíôåôáãìÝíåò. 

Ôï äïèÝí ðñüâëçìá åßíáé éóïäýíáìï ìå åîÞò: 
Íá âñåßôå ôï åìâáäü ôïõ ôìÞìáôïò ôïõ ðáñáâïëïåéäïýò z = (x^2) + (y^2) ðïõ âñßóêåôáé ìÝóá óôïí êýëéíäñï (x^2)+(y^2)=9.
ÁðëÜ èÝôïõìå x=z, z=x. 

Ãéá ôá x êáé y Ý÷ïõìå
            x= r Cos[θ]    êáé       y= r Sin[θ].  
Ãéá ôï z, ôçí ôñßôç óõíôåôáãìÝíç, Ý÷ïõìå  áðü ôçí åîßóùóç ôïõ ðáñáâïëïåéäïýò 
           z = (r Cos[θ])\(TraditionalForm\`\^2\)  + (r Sin[θ])\(TraditionalForm\`\^2\) = (r^2).  
           
Õðïëïãßæïõìå ðñþôá ôïí ÓõíôåëåóôÞÅðéöáíåßáò

\(
Clear[f, x, y, S, r, θ, J]\)
f[x_, y_] = (x^2) + (y^2);
x[r_, θ_] = r Cos[θ];
\(\(y[r_, θ_] = r\ Sin[θ];\)
\)
S[r_, θ_] = 
      SA[{x[r, θ], y[r, θ], 
            f[x[r, θ], y[r, θ]]}, {r, θ}] // 
        Simplify

Ïðüôå ôï æçôïýìåíï ïëïêëÞñùìá åßíáé

                    (∫_0^2)(∫_0^(2 π)) (Sqrt[(r^2) + 
                  4 (r^4)]) dθ dr

×ñçóéìïðïéþíôáò ôï Mathematica åýêïëá õðïëïãßæïõìå

(∫_0^3)(∫_0^(2 π)) 
        S[r, θ] \[DifferentialD]θ \[DifferentialD]r