Undergrad Transforming Vector Fields between Cylindrical Coordinates

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Transforming vector fields between cylindrical coordinate systems can be complex, especially when changing the origin and orientation of the axes. The challenge arises because cylindrical coordinates are local, meaning that unit vectors change with the origin. A recommended approach is to convert the vector field to Cartesian coordinates, apply the necessary transformations, and then convert it back to the desired cylindrical coordinate system. This method simplifies the process when the two cylindrical systems do not share an axis. Understanding these transformations is crucial for accurately analyzing rotating objects in different coordinate systems.
Luke Tan
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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!
 
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Unless the two cylindrical systems share an axis, convert to Cartesian and then back into the other cylindrical coordinate system.
 

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