« Calcul différentiel/Exercices/Différentiabilité » : différence entre les versions

Contenu supprimé Contenu ajouté
→‎Exercice 7 : +1 question
→‎Exercice 7 : calcul du laplacien en coordonnées polaires
Ligne 146 :
&=\begin{pmatrix}\frac{\partial(f\circ\phi)}{\partial r}&\frac{\partial(f\circ\phi)}{\partial\theta}\end{pmatrix}(r,\theta)\times\begin{pmatrix}\cos\theta&\sin\theta\\-\frac{\sin\theta}r&\frac{\cos\theta}r\end{pmatrix}\\
&=\begin{pmatrix}\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\cos\theta-\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\sin\theta}r&\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\sin\theta+\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\cos\theta}r\end{pmatrix}.
\end{align}</math><br>Un autre méthode est d'utiliser la sous-question précédente :<br><math>\frac{\partial f}{\partial x}(x,y)=\frac1{x^2+y^2}\left(x\Theta_f(x,y)-y\Psi_f(x,y)\right)</math> donc<br><math>\frac{\partial f}{\partial x}\circ\phi(r,\theta)=\frac1{r^2}\left(r\cos\theta\times r\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)-r\sin\theta\times\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\right)=\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\cos\theta-\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\sin\theta}r</math> ;<br><math>\frac{\partial f}{\partial y}(x,y)=\frac1{x^2+y^2}\left(y\Theta_f(x,y)+x\Psi_f(x,y)\right)</math> donc<br><math>\frac{\partial f}{\partial y}\circ\phi(r,\theta)=\frac1{r^2}\left(r\sin\theta\times r\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)+r\cos\theta\times\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\right)=\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\sin\theta+\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\cos\theta}r</math>.
\end{align}</math>
##<br><math>\begin{align}\frac{\partial^2f}{\partial x^2}\circ\phi(r,\theta)
##(en cours) [http://www.cafepedagogique.net/communautes/MarcCourbot/Documents/TRAVAUX%20SUR%20LE%20LAPLACIEN/2D%20-%20LAPLACIEN.pdf]{{Wikipédia|Opérateur laplacien}}
 
&=\frac{\partial(\frac{\partial f}{\partial x}\circ\phi)}{\partial r}(r,\theta)\cos\theta-\frac{\partial(\frac{\partial f}{\partial x}\circ\phi)}{\partial\theta}(r,\theta)\frac{\sin\theta}r\\
 
&=\frac{\partial}{\partial r}\left(\frac{\partial(f\circ\phi)}{\partial r}\cos\theta-\frac{\partial(f\circ\phi)}{\partial\theta}\frac{\sin\theta}r\right)(r,\theta)\cos\theta-\frac{\partial}{\partial\theta}\left(\frac{\partial(f\circ\phi)}{\partial r}\cos\theta-\frac{\partial(f\circ\phi)}{\partial\theta}\frac{\sin\theta}r\right)(r,\theta)\frac{\sin\theta}r\\
 
&=\frac{\partial^2(f\circ\phi)}{\partial r^2}(r,\theta)\cos^2\theta-\frac{\partial^2(f\circ\phi)}{\partial r\partial\theta}(r,\theta)\frac{\sin\theta\cos\theta}r
+\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\sin\theta\cos\theta}{r^2}\\
 
&-\frac{\partial^2(f\circ\phi)}{\partial\theta\partial r}(r,\theta)\frac{\cos\theta\sin\theta}r+\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\frac{\sin^2\theta}r+\frac{\partial^2(f\circ\phi)}{\partial\theta^2}(r,\theta)\frac{\sin^2\theta}{r^2}+\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\cos\theta\sin\theta}{r^2}\\
 
&=\frac{\partial^2(f\circ\phi)}{\partial r^2}(r,\theta)\cos^2\theta-2\frac{\partial^2(f\circ\phi)}{\partial r\partial\theta}(r,\theta)\frac{\sin\theta\cos\theta}r
+\frac{\partial^2(f\circ\phi)}{\partial\theta^2}(r,\theta)\frac{\sin^2\theta}{r^2}\\
 
&+\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\frac{\sin^2\theta}r+2\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\sin\theta\cos\theta}{r^2}
\end{align}</math><br>et<br><math>
 
\begin{align}\frac{\partial^2f}{\partial y^2}\circ\phi(r,\theta)&=\frac{\partial(\frac{\partial f}{\partial y}\circ\phi)}{\partial r}(r,\theta)\sin\theta+\frac{\partial(\frac{\partial f}{\partial y}\circ\phi)}{\partial\theta}(r,\theta)\frac{\cos\theta}r\\
 
&=\frac{\partial}{\partial r}\left(\frac{\partial(f\circ\phi)}{\partial r}\sin\theta+\frac{\partial(f\circ\phi)}{\partial\theta}\frac{\cos\theta}r\right)(r,\theta)\sin\theta+\frac{\partial}{\partial\theta}\left(\frac{\partial(f\circ\phi)}{\partial r}\sin\theta+\frac{\partial(f\circ\phi)}{\partial\theta}\frac{\cos\theta}r\right)(r,\theta)\frac{\cos\theta}r\\
 
&=\frac{\partial^2(f\circ\phi)}{\partial r^2}(r,\theta)\sin^2\theta
+\frac{\partial^2(f\circ\phi)}{\partial r\partial\theta}(r,\theta)\frac{\cos\theta\sin\theta}r
-\frac{\partial(f\circ\phi)}{\partial\theta}\frac{\cos\theta\sin\theta}{r^2}\\
 
&+\frac{\partial^2(f\circ\phi)}{\partial\theta\partial r}(r,\theta)\frac{\sin\theta\cos\theta}r
+\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\frac{\cos^2\theta}r
+\frac{\partial^2(f\circ\phi)}{\partial\theta^2}(r,\theta)\frac{\cos^2\theta}{r^2}
-\frac{\partial(f\circ\phi)}{\partial\theta}\frac{\sin\theta\cos\theta}{r^2}\\
 
&=\frac{\partial^2(f\circ\phi)}{\partial r^2}(r,\theta)\sin^2\theta
+2\frac{\partial^2(f\circ\phi)}{\partial r\partial\theta}(r,\theta)\frac{\cos\theta\sin\theta}r
+\frac{\partial^2(f\circ\phi)}{\partial\theta^2}(r,\theta)\frac{\cos^2\theta}{r^2}\\
 
&+\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)\frac{\cos^2\theta}r
-2\frac{\partial(f\circ\phi)}{\partial\theta}(r,\theta)\frac{\cos\theta\sin\theta}{r^2}
\end{align}</math><br>donc<br><math>
 
(\Delta f)\circ\phi(r,\theta)=\frac{\partial^2(f\circ\phi)}{\partial r^2}(r,\theta)+\frac1r\frac{\partial(f\circ\phi)}{\partial r}(r,\theta)
+\frac1{r^2}\frac{\partial^2(f\circ\phi)}{\partial\theta^2}(r,\theta)</math>.
{{Wikipédia|Opérateur laplacien}}
}}