<div class="exemple">
La norme du  vecteur normal en un point de paramtres \theta, \varphi est gal  \(|\cos(\varphi)|)  pour \fold{normalsphere}{la paramtrisation choisie.}

L'lment de surface est donc 
<center>\(d\Sigma=|\cos(\varphi)| d\theta d\varphi= \cos(\varphi) d\theta d\varphi) </center>

car \(\cos(\varphi)) est positif entre -\pi/2 et \pi/2.

\def{real a=random(0..pi)}
\def{real b=random(\a+0.5..2*pi)}
\def{text data=cos(u)*cos(v),sin(u)*cos(v), sin(v),\a,\b,-pi/2,pi/2,-1,1,-1,1,-1,1}
\def{text data1=\data[1]}
\def{text data2=\data[2]}
\def{text data3=\data[3]}
\def{text data4=\data[4]}
\def{text data5=\data[5]}
\def{text data6=\data[6]}
\def{text data7=\data[7]}
\def{text data8=\data[8]}
\def{text data9=\data[9]}
\def{text data10=\data[10]}
\def{text data11=\data[11]}
\def{text data12=\data[12]}
\def{text data13=\data[13]}

\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20">} 
L'aire de la
\tool{module=tool/geometry/animtrace.fr&+cmd=new&+type=parametric3DS&+special_parm=noshow&+quality=4&+x1=\data1&+y1=\data2&+z1=\data3&+uleft=\data4&+uright=\data5&+vleft=\data6&vright=\data7&
+xleft=\data8&+xright=\data9&+yleft=\data10&+yright=\data11&+zleft=\data12&+zright=\data13}{
 surface}
 sur la sphre correspondant au quartier d'orange
\(\theta_1\leq \theta \leq \theta_2, \varphi \in [-\pi/2,\pi/2]) 
est
<center>\(\int_{\theta_1}^{\theta_2} \int_{-\pi/2}^{\pi/2} cos(\varphi)d\varphi d \theta= 
2(\theta_2-\theta_1) )
.</center>
\def{real a=random(-pi/2..0.1)}
\def{real b=random(\a+0.5..pi/2)}
\def{text data=cos(u)*cos(v),sin(u)*cos(v), sin(v),0,2*pi,\a,\b,-1,1,-1,1,-1,1}
\def{text data1=\data[1]}
\def{text data2=\data[2]}
\def{text data3=\data[3]}
\def{text data4=\data[4]}
\def{text data5=\data[5]}
\def{text data6=\data[6]}
\def{text data7=\data[7]}
\def{text data8=\data[8]}
\def{text data9=\data[9]}
\def{text data10=\data[10]}
\def{text data11=\data[11]}
\def{text data12=\data[12]}
\def{text data13=\data[13]}
\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20">} 
L'aire de la 
\tool{module=tool/geometry/animtrace.fr&+cmd=new&+type=parametric3DS&+special_parm=noshow&+quality=4&+x1=\data1&+y1=\data2&+z1=\data3&+uleft=\data4&+uright=\data5&+vleft=\data6&vright=\data7&
+xleft=\data8&+xright=\data9&+yleft=\data10&+yright=\data11&+zleft=\data12&+zright=\data13}{
surface} sur la sphre correspondant 
\(\theta \in [0,2\pi], \varphi \in [\varphi_1,\varphi_2]) 
est
<center>\(\int_{0}^{2\pi} \int_{\varphi_1}^{\varphi_2} cos(\varphi)d\varphi d \theta= 
2\pi sin(\varphi_2)-\sin(\varphi_1) ).</center>

\def{real a=random(-pi/2..0.1)}
\def{real b=random(\a+0.5..pi/2)}

\def{real a1=random(0..pi)}
\def{real b1=random(\a+0.5..2*pi)}

\def{text data=cos(u)*cos(v),sin(u)*cos(v), sin(v),\a1,\b1,\a,\b,-1,1,-1,1,-1,1}
\def{text data1=\data[1]}
\def{text data2=\data[2]}
\def{text data3=\data[3]}
\def{text data4=\data[4]}
\def{text data5=\data[5]}
\def{text data6=\data[6]}
\def{text data7=\data[7]}
\def{text data8=\data[8]}
\def{text data9=\data[9]}
\def{text data10=\data[10]}
\def{text data11=\data[11]}
\def{text data12=\data[12]}
\def{text data13=\data[13]}
\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20">} 
L'aire de la 
\tool{module=tool/geometry/animtrace.fr&+cmd=new&+type=parametric3DS&+special_parm=noshow&+quality=4&+x1=\data1&+y1=\data2&+z1=\data3&+uleft=\data4&+uright=\data5&+vleft=\data6&vright=\data7&
+xleft=\data8&+xright=\data9&+yleft=\data10&+yright=\data11&+zleft=\data12&+zright=\data13}{
surface} sur la sphre correspondant 
\(\theta \in [\theta_1,\theta_2], \varphi \in [\varphi_1,\varphi_2]) 
est
<center>\(\int_{-\theta_1}^{\theta_2} \int_{\varphi_1}^{\varphi_2} cos(\varphi)d\varphi d \theta= 
(\theta_2-\theta_1)(sin(\varphi_2)-\sin(\varphi_1)) ).</center>


Par exemple l'aire de la sphre de rayon 1 est \(4*pi). 
</div>