Question 2
Durée : 5 mn
Note maximale : 7
Question
Calculer la dérivée partielle \(\frac{\delta A}{\delta n}.\)
\(\sin i = n\sin r~~~~~\color{blue}(1)\\\color{black}\sin i' = n\sin r' ~~\color{blue}(2)\\\color{black}r+r'=A~~~~~~~~~~\color{blue}(3)\\\color{black}D=i+i'-A~~~~\color{blue}(4)\)
\(n,\) indice du prisme
Solution
Pour \(A\) et \(i\) constants, la différentiation des formules du prisme conduit à :
\(\color{blue}\begin{array}{r c l c}0&=&n\cos r~dr+\sin r~dn&~~\color{red}\textrm{(1 point)}\\\cos i'di'&=&n\cos r'dr'+\sin r' dn&~~\color{red}\textrm{(1 point)}\\0&=&dr+dr'&~~\color{red}\textrm{(1 point)}\\dD&=&di&~~\color{red}\textrm{(1 point)}\end{array}\)
la résolution donne
\(\color{blue}\frac{\delta D}{\delta n}=\frac{\sin A}{\cos r \cos i'}~~\color{red}\textrm{(3 points)}\)
avec \(\sin A=\sin r\cos r'+\cos r\sin r'.\)
\(\sin i = n\sin r~~~~~\color{blue}(1)\\\color{black}\sin i' = n\sin r' ~~\color{blue}(2)\\\color{black}r+r'=A~~~~~~~~~~\color{blue}(3)\\\color{black}D=i+i'-A~~~~\color{blue}(4)\)
\(n,\) indice du prisme