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In an area called transformation optics, they transform Maxwell equations from one space coordinate system to another, and then using the fact that Maxwell equations retain the same format under coordinate transformations in space, and the fact that Maxwell equations interpreted in different coordinate systems are equivalent to changing the medium parameters $(\epsilon , \mu)$ in the constitutive equations, obtain the properties of background material $(\epsilon , \mu)$ in the first coordinate system, and this find the required $(\epsilon , \mu)$ to direct EM waves in arbitrary directions.

See these papers for more information (not necessary for this question): General relativity in electrical engineering

From this paper, we have a transformation defined in cylindrical coordinates as : $$\rho'=R_1+\frac{R_2-R_1}{R_2}\rho$$ $$\phi'=\phi$$ $$z'=z$$

This transformation maps the circular region $0\leq\rho \leq R$ to the circular annular region $R_1\leq \rho' \leq R_2$. In the paper, the Jacobi matrix for this transformation is written as (equation 14): $$ A=\begin{pmatrix} (R_2-R_1)/R_2 & 0 & 0\\ 0 & \rho'/\rho & 0\\ 0& 0 & 1 \end{pmatrix}$$

We know that A is defined as $$ A=\begin{pmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial y} & \frac{\partial x'}{\partial z}\\ \frac{\partial y'}{\partial x} & \frac{\partial y'}{\partial y} & \frac{\partial y'}{\partial z}\\ \frac{\partial z'}{\partial x} & \frac{\partial z'}{\partial y} & \frac{\partial z'}{\partial z} \end{pmatrix}$$

Why the element $A_{22}=\rho'/\rho$ instead of $1$? (This result is obtained in equation (11) in the above paper, using more accurate mathematical notation.)

In another transformation, defined as:

$$\rho'=\rho$$ $$\phi'=\frac{a}{2l}(z+l)+\phi$$ $$z'=z$$ The matrix $A$ is obtained as:

$$ A=\begin{pmatrix} 1 & 0 & 0\\ 0 & 1& \frac{a}{2l}\rho'\\ 0& 0 & 1 \end{pmatrix}$$

In this example, why the $A_{23}$ element is not $\frac{a}{2l}$?

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