Proper change of coordinates It's a really easy one for you guys.
I'm performing a simple cylindrical change of variable from Cartesian coordinates, but I want to write it out properly and I'm stuck with the differential matrices. I won't get into details on everything here but I'll focus on my issue.
I consider the diffeomorphism $\Psi$ matching the cylindrical change of coordinates $\Psi : (r, \phi, z) \rightarrow (r\cos{\phi}, r\sin{\phi}, z)$
And I compute its jacobian $D\Psi$ as well as its inverse $(D\Psi)^{-1}$.
Now, for any vector field $A : (x, y, z) \rightarrow (A_x(x, y, z), A_y(x, y, z), A_z(x, y, z))$ and $\hat{A}  : (r, \phi, z) \rightarrow (A_r(r, \phi, z), A_\phi(r, \phi, z), A_z(r, \phi, z))$ I would like to set clearly the link between partial derivatives of these two vector fields in a matrix form such that, $D\hat{A}(r, \theta, z) = DA(\Psi(r, \phi, z))\circ D\Psi(r, \phi, z)$,
(edit) Where $D$ is the jacobian matrix.
My final goal would be to get $\partial_x A$ in function of the partial derivatives $\partial_r, \partial_\phi, \partial_z$ of $\hat{A}$ in order to express the differential operators such as curl or divergence in the new coordinate system, but I'm not sure how to do it. Thus, could someone help me out to clarify the procedure of this change of coordinates.
(edit) A confusing point I think is that we need to be cautious about the basis in which matrices are expressed. This is why I have trouble apprehending the matrix formulation, I don't know if we can get something like $J(A)_{x, y, x} = T J(\hat{A})_{r, \phi, z}$ where $J$ is the jacobian, and $T$ a matrix to be computed.
I'm a bit ashamed to ask it here, but sometimes you just can't figure out basic stuff.
Thanks in advance.
 A: I'm not as familiar with the Jacobian-based methods, although all methods for answering this are equivalent. Here's what I would do.
Ignore the $z$-coordinate, since it is the same for both coordinate systems, and because coordinate basis vector fields are linearly independent (i.e., if $e_x,e_y,e_z$ is the Cartesian basis then the vector fields that form it are linearly independent; the same holds for any other basis; however if $e_r,e_\phi,e_z$ is the polar basis then there can be a relationship between, e.g., $e_r,$ $e_x$ and $e_y$). Therefore just look at polars $r,\phi$ in the $x,y$-plane. Write down the conversion formulae (you already have them implicitly in your diffeo):
\begin{align} x   &= r \cos (\phi), \\
              y   &= r \sin (\phi). \end{align}
This gives
\begin{align} r    &= (x^2 + y^2)^{1/2}, \\
              \phi &= \tan^{-1} (\frac{y}{x}). \end{align}
Hence we have $\partial r / \partial y = \frac{y}{(x^2 + y^2)^{1/2}} = \sin(\phi)$ and $\partial \phi / \partial y = \frac{x}{x^2 + y^2} = \frac{\cos(\phi)}{r}$. You should find that these two results are exactly two of the terms in the Jacobian of the transformation (similarly for the derivatives w.r.t. $x$). The chain rule of multi-variable calculus then gives that, e.g.,
\begin{align} \frac{\partial}{\partial y} &= \frac{\partial r}{\partial y} \frac{\partial}{\partial r} + \frac{\partial \phi}{\partial y} \frac{\partial}{\partial \phi} \\
&= \sin(\phi) \frac{\partial}{\partial r} + \frac{\cos(\phi)}{r} \frac{\partial}{\partial \phi}. \end{align}
This is, I believe, the operator you sought in your final answer; you can then apply this to an arbitrary vector field, ideally given using the same cylindrical polar coordinates.
