Given is a coordinate system $\{x_1,x_2,...,x_n\}$ and another, second coordinate system $\{y_1,y_2,...,y_n\}$, where
$x_1=x_1(y_1,y_2,...,y_n)$
$x_2=x_2(y_1,y_2,...,y_n)$
...
$x_n=x_n(y_1,y_2,...,y_n)$
Then the Jacobian matrix is
$${\mathbf J}=\begin{pmatrix} \frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2} & ... & \frac{\partial x_1}{\partial y_n}\\ \frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2} & ... & \frac{\partial x_2}{\partial y_n}\\ ... \\ \frac{\partial x_n}{\partial y_1} & \frac{\partial x_n}{\partial y_2} & ... & \frac{\partial x_n}{\partial y_n}\\ \end{pmatrix}$$
Now, consider vector $\vec{u}$. Its coordinates in $\{x_k\}$ are $\vec{u}=(u_{x1},u_{x2},...,u_{xn})$, while its coordinates in $\{y_k\}$ are $\vec{u}=(u_{y1},u_{y2},...,u_{yn})$, where
$$\begin{pmatrix} u_{x1}\\ u_{x2}\\ ... \\ u_{xn}\\ \end{pmatrix}= {\mathbf A} \cdot \begin{pmatrix} u_{y1}\\ u_{y2}\\ ... \\ u_{yn}\\ \end{pmatrix}$$
What is the difference between the Jacobian matrix ${\mathbf J}$ and the Transformation matrix ${\mathbf A}$? How are they related? Please, write the expression that connects them.
------------EXAMPLE-----------
Cartesian and cylindrical coordinates are related via
$x=r\cos\theta$
$y=r\sin\theta$
$z=z$
Then the Jacobian is
$${\mathbf J}=\begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z}\\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z}\\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z}\\ \end{pmatrix}= \begin{pmatrix} \cos\theta & -r\sin\theta & 0\\ \sin\theta & r\cos\theta & 0\\ 0 & 0 & 1\\ \end{pmatrix}$$
and $\det({\mathbf J})=r$. Because ${\mathbf J}$ is orthogonal when $r=1$,
${\mathbf J}^{-1}={\mathbf J}^{T}=\begin{pmatrix} \cos\theta & \sin\theta & 0\\ -\sin\theta & \cos\theta & 0\\ 0 & 0 & 1\\ \end{pmatrix}$
The Cartesian basis vectors are
$\hat{\mathbf i} = \begin{pmatrix} 1\\ 0\\ 0\\ \end{pmatrix}$; $\hat{\mathbf j} = \begin{pmatrix} 0\\ 1\\ 0\\ \end{pmatrix}$; $\hat{\mathbf k} = \begin{pmatrix} 0\\ 0\\ 1\\ \end{pmatrix}$;
The cylindrical basis vectors are
$\hat{\mathbf r}= \begin{pmatrix} \cos\theta\\ \sin\theta\\ 0\\ \end{pmatrix}$; $\hat{\mathbf \theta}= \begin{pmatrix} -\sin\theta\\ \cos\theta\\ 0\\ \end{pmatrix}$; $\hat{\mathbf z} = \begin{pmatrix} 0\\ 0\\ 1\\ \end{pmatrix}$.
It seems that
$\hat{\mathbf r}={\mathbf A}\cdot \hat{\mathbf i}$
$\hat{\mathbf \theta}={\mathbf A}\cdot \hat{\mathbf j}$
$\hat{\mathbf z}={\mathbf A}\cdot \hat{\mathbf k}$
if the transformation matrix ${\mathbf A}$ is given by
${\mathbf A}=\frac{1}{\det({\mathbf J})}{\mathbf J}^{-1}$.
Is this conclusion true? Is this the relationship between ${\mathbf A}$ and ${\mathbf J}$?
