\def{integer n=randint(2..4)}
<div class="defn">Soit  \(f: {\mathcal U}\subset \RR^\n\to \RR) 
\reload{<img src="gifs/doc/etoile.gif" alt="rechargez" width="20" height="20">} 
une fonction de \n variables. On lui
associe un champ de vecteurs appel <span class="defn">champ de gradient </span>
et not grad \( f) ou \nabla  \(f) :<center> 
\if{\n=2}{\(
(x,y)\mapsto \nabla  f (x,y)= (\frac{\partial f}{\partial x}(x,y),\frac{\partial
f}{\partial y}(x,y))
\)
</center>}
\if{\n=3}{ \(
(x,y,z)\mapsto \nabla  f (x,y,z)= (\frac{\partial f}{\partial x}(x,y,z),\frac{\partial
f}{\partial y}(x,y,z),\frac{\partial
f}{\partial z}(x,y,z)))
</center>}
\if{\n>3}{\def{text liste=x<sub>1</sub>}
\for{i=2 to \n}{\def{text liste=\liste, x<sub>\i</sub>}
\def{text listpart=D<sub>1</sub> f(\liste)}
}
\def{text listpart2=D<sub>1</sub> f(M)}
\def{text listpart1=D<sub>1</sub> f e<sub>1</sub>}
\for{i=2 to \n}{\def{text listpart=\listpart, D<sub>\i</sub> f(\liste)}
\def{text listpart2=\listpart2, D<sub>\i</sub> f(M)}
\def{text listpart1=\listpart1  +  D<sub>\i</sub> f e<sub>\i</sub>}
}
(\liste)\mapsto \nabla  f (\liste)= (\listpart)
</center> avec \(D_i= \frac{\partial f}{\partial x_i}).
}</div>

En posant \if{\n=2}{\(M=(x,y))}\if{\n=3}{\(M=(x,y,z))}\if{\n>3}{\(M)=(\liste)}, 
<center>  \if{\n=2}{grad \(f(M)=(\frac{\partial f}{\partial x}(M),\frac{\partial
f}{\partial y}(M))).
}
\if{\n=3}{grad \(f(M)=(\frac{\partial f}{\partial x}(M),\frac{\partial
f}{\partial y}(M),\frac{\partial
f}{\partial z}(M))).
}
\if{\n>3}{grad \(f(M))=(\listpart2).}
</center>
 
 <div class="exercice"> \exercise{cmd=new&module=U2/analysis/oefchamp.fr&exo=champgrad}{<span class="exercice">Exercice</span>}
</div>

<div class="defn"><span class="definition"> Autres notations :  </span>
<ul>
<li> en utilisant la base  canonique  (\for{i=1 to \n}{
	\def{text temp=e<sub>\i</sub>} \if{\i<\n}{\temp,}{\temp}})

<center> 
 \nabla f = \if{\n=2}{
 	\(\frac{\partial f}{\partial x} e_1+ \frac{\partial f}{\partial y}e_2)}
 	\if{\n=3}{
 		\( \frac{\partial f}{\partial x} e_1+ 
 		 \frac{\partial f}{\partial y}e_2+\frac{\partial f }{\partial z}e_3)
 	}
 	\if{\n>3}{\listpart1}
 	</li>
 	<li>	
En physique, on  utilise la  notation suivante : 
 \(u_x=e_1),
 \(u_y=e_2),  \(u_z=e_3) ce qui donne les formules suivantes 
<center> \(
 \nabla = \frac{\partial }{\partial x} u_x+ \frac{\partial }{\partial y}u_y+
\frac{\partial }{\partial z}u_z) dans \(\RR^3)
<br> \(
 \nabla = \frac{\partial }{\partial x} u_x+ \frac{\partial }{\partial y}u_y)
 dans \(\RR^2) 
 </center>
ou en mettant les scalaires aprs les vecteurs contrairement  nos habitudes
<center> \(
 \nabla =  u_x\frac{\partial }{\partial x}+  u_y\frac{\partial }{\partial y}+
 u_z\frac{\partial }{\partial z}) dans \(\RR^3)
<br> \(
 \nabla =  u_x\frac{\partial }{\partial x}+  u_y\frac{\partial }{\partial y})
 dans \(\RR^2) .
 </center>
 </li>
</ul>
</div>