Understanding Metamaterials Equations

[JasonGodbout]

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

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

$M = \begin{bmatrix} \frac{B_{r0}}{\mu_{0}} & r' \epsilon, [0 , R_1] \\ 0 & r' \epsilon [R_1 , R_3] \\ 0 & r' \epsilon [R_3 , \infty \end{bmatrix}\\$

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 :
$M' = \begin{bmatrix} diag( \frac{R_{2}}{R_{1}}, \frac{R_{2}}{R_{1}}, (\frac{R_{2}}{R_{1}})^2)\frac{B_{r0}}{\mu_{0}} & r' \epsilon, [0 , R_1] \\ 0 & r' \epsilon [R_1 , R_3] \\ 0 & r' \epsilon [R_3 , \infty \end{bmatrix}\\$

 

[mighty2000]

Hi there! It seems like you are having trouble understanding the metamaterials equation and the role of the Jacobian matrix. Let me try to explain it in simpler terms.

The metamaterials equation is used to describe the behavior of electromagnetic waves in a material that has been engineered to have unique properties. In this equation, the matrix M represents the material's response to an incident electromagnetic wave, and M' represents the material's response to the same wave after it has been transformed by a certain transformation matrix.

The transformation matrix, denoted by Λ', is used to describe the change in coordinates from the original coordinate system (r, θ, z) to the transformed coordinate system (r', θ', z'). The Jacobian matrix, denoted by Λ, is used to calculate the transformation matrix Λ' and takes into account the change in the coordinates due to the transformation.

In your attempt at a solution, you correctly identified the Jacobian matrix Λ' as a 3x3 matrix. However, the transformation matrix Λ' should also be a 3x3 matrix, and not a 3x1 matrix as you have written. This is because the transformation matrix should have the same number of rows and columns as the Jacobian matrix.

To calculate the determinant of the transformation matrix, you need to take the determinant of each sub-matrix within the transformation matrix. In this case, there are three sub-matrices, one for each region of r' (0<r'<R1, R1<r'<R3, r'>R3). For each sub-matrix, you would use the corresponding values from the Jacobian matrix to calculate the determinant.

Once you have calculated the determinant of the transformation matrix, you can use it in the metamaterials equation to find the transformed matrix M'. I hope this helps clarify the process for you. Keep up the good work in understanding metamaterials!