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Jacobian Matrix

  1. Jul 8, 2017 #1
    1. The problem statement, all variables and given/known data
    I am trying to understand Metamaterials eqaution but the matrix always have mutiple r'(r) let use an example I saw on the web, this on his in cylindrical coordinate:

    ##M^{i'} = \frac{1}{det\Lambda'}\Lambda_{i}^{i'}M^{i} ##

    ##\Lambda'## is de Jacobien
    2. Relevant equations

    ##r' =
    \begin{bmatrix}
    \frac{R_{1}r}{R_{2}} & r' \epsilon, [0 , R_1] \\
    \frac{(R_{3}-R_{1})r}{R_{3}-R_{2}}+\frac{R_{3}(R_{1}-R_{2})}{R_{3}-R_{2}} & r' \epsilon [R_1 , R_3] \\
    r & r' \epsilon [R_3 , \infty
    \end{bmatrix}\\##

    ##M =
    \begin{bmatrix}
    \frac{B_{r0}}{\mu_{0}} & r' \epsilon, [0 , R_1] \\
    0 & r' \epsilon [R_1 , R_3] \\
    0 & r' \epsilon [R_3 , \infty
    \end{bmatrix}\\##
    3. The attempt at a solution

    ##Trasformation Matrix =
    \begin{bmatrix}
    \begin{bmatrix}\frac{R_{1}r}{R_{2}} & \frac{(R_{3}-R_{1})r}{R_{3}-R_{2}}+\frac{R_{3}(R_{1}-R_{2})}{R_{3}-R_{2}} & r\end{bmatrix} & 0 & 0 \\
    0 & \theta & 0 \\
    0 & 0 & z
    \end{bmatrix}
    ##
    I guess the Jacobian is :
    ##\Lambda' =
    \begin{bmatrix}
    \begin{bmatrix}\frac{R_{1}}{R_{2}} & \frac{R_{3}-R_{1}}{R_{3}-R_{2}} & 1\end{bmatrix} & 0 & 0 \\
    0 & 1 & 0 \\
    0 & 0 & 1
    \end{bmatrix}\\
    ##

    but then I get 3 determinant (0<r'<R1, R1<r'<R3, r'>R3) and I don't know how to get the answer :
    ##M' =
    \begin{bmatrix}
    diag( \frac{R_{2}}{R_{1}}, \frac{R_{2}}{R_{1}}, (\frac{R_{2}}{R_{1}})^2)\frac{B_{r0}}{\mu_{0}} & r' \epsilon, [0 , R_1] \\
    0 & r' \epsilon [R_1 , R_3] \\
    0 & r' \epsilon [R_3 , \infty
    \end{bmatrix}\\
    ##
     
    Last edited: Jul 9, 2017
  2. jcsd
  3. Jul 14, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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