Jacobian Matrix

  • #1

Homework Statement


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

Homework Equations


[/B]
##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}\\##

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:

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