<div class="exemple">Prenons pour \calS une surface dcrite par une quation explicite \(z = g(x,y)\).
Soit 
\( F = (P , Q , R) ) un champ sur \calS, alors

 <p> <center>\(\int\!\!\int_{\mathcal S} F \cdot\vec{dS}= \int\!\!\int_{\mathcal D} (R-PD_1(g) -Q D_2(g))dx dy=
\int\!\!\int_{\mathcal D} F\cdot \begin{pmatrix}-grad(g)\\1\end{pmatrix}dx dy
 )
\(=\int\!\!\int_{\mathcal D} \begin{pmatrix}P\\Q\\R\end{pmatrix}\cdot \begin{pmatrix}-\frac{\partial g}{\partial x}\\-\frac{\partial g}{\partial y}\\1\end{pmatrix}dx dy) </center></p>

</div>

<div class="exemple">Prenons pour \calS une portion de sphre unit : \(\theta\in [\theta_1,\theta_2]), \(\varphi\in [\varphi_1,\varphi_2])
Soit 
\( F = (P , Q , R) ) un champ sur \calS, alors
<center>\(\int\!\!\int_{\mathcal S} F \cdot\vec{dS})=
=
\(\int_{\theta_1}^{\theta_2}\int_{\varphi_1}^{\varphi_2}
 \vec{OM}\cdot F(M)\cos(\varphi)d\varphi
d\theta
)</center>
<center>

=
 \(\int_{\theta_1}^{\theta_2}\int_{\varphi_1}^{\varphi_2}
 \left (x(\theta,\varphi)P(M(\theta,\varphi))) \right .)
+  \(
y(\theta,\varphi)Q(M(\theta,\varphi))  ) + \(\left .z(\theta,\varphi)R(M(\theta,\varphi)\right )\cos(\varphi)d\varphi
d\theta
)
</center><center>
=
\(\int_{\theta_1}^{\theta_2}\int_{\varphi_1}^{\varphi_2}
 \left (x P(x,y,z) 
+  
yQ(x,y,z)  +z R(x,y,z)\right )\cos(\varphi)d\varphi
d\theta
)
</center></p>
</div>