Say I have a field $\vec{F}(x,y,z) = A_r \hat{r} +A_\theta\hat{\theta} + A_z\hat{z}$. I'd like to change this field into both spherical and cartesian coordinates.
I've seen quite a few wikipedia pages with formulas like the image https://i.stack.imgur.com/CYceF.png
But I'm not exactly sure how I can use these formulas to do the conversion. For starters, $\vec{F}$ should have basis vectors $\hat{\rho}, \hat{\theta}, \hat{\phi}$, so somehow I'd have to get rid of $\hat{r}, \hat{\theta}, \hat{z}$. I'd also have to change each of $A_r, A_\theta, A_z$ into their respective spherical coordinates representations, meaning they can only contain $\rho, \theta, \phi$.
I read some example in a wiki page where transformation matrices were used to accomplish the task. These matrices can be used to turn any vector field in some coordinate system into any other coordinate system. The process is simple and robotic: it always yields the correct answer (as opposed to using intuition on a vector field, like saying $x \hat{x} + y\hat{y} + z\hat{z} = \rho\hat{\rho}$). The conversion is complete once the matrix-vector multiplication has completed. I've sadly misplaced the wiki page, and all I have left is https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates#Non-trivial_calculation_rules, where there are no transformation matrices.
Can anyone lead me to these transformation matrices, or another source that specifies how to convert vector fields mundanely? I'm not sure I can just rely on the intuitions mentioned above. I'd like a way to do it for ANY vector field, from one coordinate system to any other coordinate system.