\def{text color1=red}
\def{text color2=blue}
\def{text dessin1=
xrange 0,3
yrange 0,2
text black,1,1,medium,a
text black,2,1,medium,b
linewidth 3
line 1,1,2,1,\color2
disk  1,1,7,\color1
disk 2,1,7,\color1}

\def{text data= cos(t),t, 0,2*pi}
\def{text data=\data, evalue(\data[1],t=\data[3]), evalue(\data[2],t=\data[3]),
 evalue(\data[1],t=\data[4]), evalue(\data[2],t=\data[4])}
\def{text dessin2=
xrange -1,2
yrange -2,7
text black,\data[5],\data[6],medium,A
text black,\data[7],\data[8],medium,B
disk  \data[5],\data[6],10,\color1
disk \data[7],\data[8],10,\color1
linewidth 3
trange \data[3],\data[4]
plot \color2,  \data[1],\data[2]

}
<table align="center" border=1>
<tr align=center><td> <font color=\color2>courbe</font> borde par des <font color=\color1>points</font> dans \(\RR)</td>
<td>\(\special{color=\color2}\int_{ a  }^{b }\special{color=black}f'(x) dx)</td>
<td>=</td>
<td>\(f(\special{color=\color1}b\special{color=black})-f(\special{color=\color1}a\special{color=black}))</td>
<td> \draw{200,100}{\dessin1}</td>
<td>\(\partial \special{color=\color2}C\special{color=black}=\special{color=\color1}\lbrace a,b \rbrace\special{color=black})</td>
</tr>
<tr align=center><td> <font color=\color2>courbe</font> borde par des <font color=\color1>points</font> dans \(\RR^2)</td>
<td>
\(\special{color=\color2}\int_C \special{color=black}{\vec{\rm grad
}}\  f(M) \cdot \vec{dM})</td>
<td>=</td>
<td>\(f(\special{color=\color1}B\special{color=black})-f(\special{color=\color1}A\special{color=black}))</td>
<td> \draw{100,100}{\dessin2}</td>
<td>\(\partial \special{color=\color2}C\special{color=black}=\special{color=\color1}\lbrace A,B\rbrace\special{color=black}))</td>
</tr>
<tr align=center><td> <font color=\color2>courbe</font> sans (points) bord dans \(\RR^2)</td>
<td>
\(\special{color=\color2}\int_C \special{color=black}{\vec{\rm grad
}}\  f(M) \cdot \vec{dM}) </td>
<td>=</td>
<td>0 
</td>
\def{text data= cos(t),sin(t), 0,2*pi}
\def{text data=\data, evalue(\data[1],t=\data[3]), evalue(\data[2],t=\data[3]),
 evalue(\data[1],t=\data[4]), evalue(\data[2],t=\data[4])}
\def{text dessin3=xrange -2,2
yrange -2,2
linewidth 2
trange \data[3],\data[4]
plot \color2,  \data[1],\data[2]
}
<td> \draw{100,100}{\dessin3}</td>
<td>\(\partial \special{color=\color2}C\special{color=black}=\emptyset)</td>
</tr>
<tr align=center> <td> <font color=\color2>domaine</font>  bord par une <font color=\color1>courbe</font> dans \(\RR^2)</td>
<td>
\(\special{color=\color2}\int\!\!\int_D \special{color=black}  {\rm rot} \  \vec{F}(M) \ dA)</td>
<td>=</td>
<td> \(\special{color=\color1}\int_C \special{color=black}  \vec{F}(M)\cdot \vec{dM}) 
</td>
\def{text data= cos(t),sin(t), 0,2*pi}
\def{text data=\data, evalue(\data[1],t=\data[3]), evalue(\data[2],t=\data[3]),
 evalue(\data[1],t=\data[4]), evalue(\data[2],t=\data[4])}
\def{text dessin4=xrange -2,2
yrange -2,2
text black,\data[5],\data[6],medium,A
text black,\data[7],\data[8],medium,B
linewidth 2
trange \data[3],\data[4]
plot \color1,  \data[1],\data[2]
fill 0,0, \color2
arrow cos(pi/3),sin(pi/3),cos(pi/3)-sin(pi/3),sin(pi/3)+cos(pi/3), 8,black
}
<td> \draw{100,100}{\dessin4}</td>
<td>\(\partial \special{color=\color2}D\special{color=black}=\special{color=\color1}C\special{color=black}), \(\partial \special{color=\color1}C\special{color=black}=\emptyset)</tr>
<tr align=center><td> <font color=\color2>domaine</font>  bord par une <font color=\color1>courbe</font> dans \(\RR^2)</td>
<td>
\(\special{color=\color2}\int\!\!\int_D \special{color=black}  {\rm div} \  \vec{F}(M) \ dA)</td>
<td>=</td>
<td> \(\special{color=\color1}\int_C \special{color=black}  \vec{F}(M)\cdot \vec{dN}) 
</td>
\def{text data= cos(t),sin(t), 0,2*pi}
\def{text data=\data, evalue(\data[1],t=\data[3]), evalue(\data[2],t=\data[3]),
 evalue(\data[1],t=\data[4]), evalue(\data[2],t=\data[4])}
\def{text dessin4=xrange -2,2
yrange -2,2
linewidth 2
trange \data[3],\data[4]
plot \color1,  \data[1],\data[2]
fill 0,0, \color2
arrow  cos(pi/3-0.2),sin(pi/3-0.2),2*cos(pi/3-0.2),2*sin(pi/3-0.2), 8,black
text black, 1.5*cos(pi/3-0.2),1.5*sin(pi/3-0.2),medium, N
}
<td> \draw{100,100}{\dessin4}</td>
<td>\(\partial \special{color=\color2}D\special{color=black}=\special{color=\color1}C\special{color=black}) &nbsp; \(\partial \special{color=\color1}C\special{color=black}=\emptyset)</td></tr>

<tr align=center><td> <font color=\color2>surface</font> borde par une <font color=\color1>courbe</font> dans \(\RR^3)</td>
<td>
\(\special{color=\color2}\int\!\!\int_S\special{color=black}{\vec{\rm rot}} \   \vec{F}(M) \cdot \vec{dS})</td>
<td>=</td>
<td> \(\special{color=\color1}\int_C \special{color=black} \vec{F}(M)\cdot \vec{dM}) 
</td>
\def{text dessin6=xrange -1.5,1.5
yrange -1.5,1.5
trange -pi,0
plot \color2, cos(t),sin(t)
ellipse 0,0,2,0.5 , \color1
fill 0,-0.75,\color2
linewidth 4
ellipse 0,0,2,0.5 , \color1}
<td>
\draw{100,100}{\dessin6}
</td>
<td>\(\partial \special{color=\color2}S\special{color=black}=\special{color=\color1}C\special{color=black}) &nbsp; \(\partial \special{color=\color1}C\special{color=black}=\emptyset)</td>

</tr>
<tr align=center><td> <font color=\color2>surface</font> sans bord dans \(\RR^3)</td>
<td>
\(\special{color=\color2}\int\!\!\int_S\special{color=black}{\vec{\rm rot}} \   \vec{F}(M) \cdot \vec{dS})</td>
<td>=</td>
<td> 0
</td>
\def{text dessin7=xrange -1.5,1.5
yrange -1.5,1.5
trange 0,2*pi
plot \color2, cos(t),sin(t)
fill 0,-0.75,\color2
ellipse 0,0,2,0.5 ,black
}
<td>
\draw{100,100}{\dessin7}
</td>
<td>\(\partial \special{color=\color2}D\special{color=black}=\emptyset) &nbsp; </td>
</tr>
<tr align=center><td> <font color=\color2>volume</font> bord par une <font color=\color1>surface</font> dans \(\RR^3)</td>
<td>
\(\special{color=\color2}\int\!\!\int\!\!\int_V \special{color=black} {\rm div} \   \vec{F}(M) \ dV)</td>
<td>=</td>
<td> \(\special{color=\color1}\int\!\!\int_S \special{color=black} \vec{F}(M) \cdot \vec{dS} )
</td>
<td>\def{text dessin8=xrange -1.5,1.5
yrange -1.5,1.5
trange 0,2*pi
plot \color1, cos(t),sin(t)
fill 0,-0.75,\color1
ellipse 0,0,2,0.5 ,black

}

\draw{100,100}{\dessin8}</td>
<td>\(\partial \special{color=\color2}V\special{color=\black}=\special{color=\color1}S\special{color=black}) &nbsp;\(\partial \special{color=\color1}S\special{color=black}=\emptyset) &nbsp; </td>
</tr>
<tr align=center><td> <font color=\color2>volume</font> bord par une <font color=\color1>surface</font> dans \(\RR^3)</td>
<td>
\(\special{color=\color2}\int\!\!\int\!\!\int_V \special{color=black} {\rm \Delta}(f(M) )\ dV)</td>
<td>=</td>
<td> \(\special{color=\color1}\int\!\!\int_S \special{color=black} {\vec {\rm grad}} \ f(M)\cdot \vec{dS})
</td>
<td>
\draw{100,100}{\dessin8}</td>
<td>\(\partial \special{color=\color2}V\special{color=black}=\special{color=\color1}S\special{color=black}) &nbsp;\(\partial \special{color=\color1}S\special{color=black}=\emptyset)</td>
</tr>

</table>

Pour un volume sans bord, allez faire un tour dans \(\RR^4) ! et je n'ai pas pu reprsenter la boule 
 l'intrieur de la sphre...


<table border=1><tr align=center><td>
lment d'aire dans \(\RR^2) 
</td><td>\(dA) </td><td> \(dxdy)</td><td>\(r dr d\theta)</td></tr>
<tr align=center><td>
lment de volume dans \(\RR^3) 
</td><td>\(dV) </td><td> \(dxdydz)</td><td>\(r dr d\theta dz)</td>
<td>\(\cos(\varphi) r^2 dr d\theta d \varphi)
(\(\varphi) l'angle de \(\vec{OM}) avec \(xOy))</td>
</tr>
<tr align=center><td>
lment curviligne  dans \(\RR^3) 
</td><td>\(\vec{dM}) </td><td> \((dx,dy,dz))</td>
</tr>
<tr align=center><td>
lment de longueur  sur une courbe 
</td><td>\(\vert\vert \vec{dM}\vert \vert ) </td><td> \(||\vec{ M'(t)}||dt )</td>
</tr>
<tr align=center><td>
lment d'aire  dans \(\RR^3) 
</td><td>\(d\Sigma=||\vec{N} || du dv ) </td><td> </td>
</tr>
<tr align=center><td>
lment de surface  dans \(\RR^3) 
</td><td>\(\vec{dS}=\vec{N} \  du dv ) </td><td> </td>
</tr>
</table>