# Transforming Vector Fields between Cylindrical Coordinates

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Summary:
I would like to transform a vector field from cylindrical coordinates with one origin to cylindrical coordinates with another
In dealing with rotating objects, I have found the need to be able to transform a vector field from cylindrical coordinate systems with one set of coordinate axes to another set.

For eg i'd like to transform a vector field from being measured in a set of cylindrical coordinates with origin at (0,0,0) and z axis pointing along (0,0,1) to a set of axes with the origin at (0,1,3) and the z axis pointing along (1,1,0)

In cartesian coordinates this would be quite easy, representing it as the translation of the origin then a linear transformation. However, this is much more difficult in cylindrical coordinates due to the coordinate system being local and thus the unit vectors also change when the origin is changed. Are there any general ways to do this?

Thanks!