Transforming Vector Fields between Cylindrical Coordinates

In summary, the conversation discusses the need to transform a vector field from cylindrical coordinate systems with one set of axes to another set. This can be done easily in cartesian coordinates, but is more difficult in cylindrical coordinates due to the local nature of the coordinate system. The suggested approach is to convert to cartesian coordinates and then back into the desired cylindrical coordinate system.
  • #1
Luke Tan
29
2
TL;DR 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!
 
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  • #2
Unless the two cylindrical systems share an axis, convert to Cartesian and then back into the other cylindrical coordinate system.
 

1. What are cylindrical coordinates?

Cylindrical coordinates are a system of coordinates used to describe points in three-dimensional space. They consist of a radial distance from a central axis, an angle from a reference plane, and a height or depth from a reference plane.

2. Why do we need to transform vector fields between cylindrical coordinates?

Transforming vector fields between cylindrical coordinates allows us to analyze and manipulate vector quantities, such as velocity or force, in a more convenient and efficient way. It also helps us to better understand and visualize the behavior of these vector fields in different coordinate systems.

3. How do you convert a vector field from cylindrical coordinates to Cartesian coordinates?

To convert a vector field from cylindrical coordinates to Cartesian coordinates, we use the following equations:

x = r * cos(θ)

y = r * sin(θ)

z = z

Where r is the radial distance, θ is the angle, and z is the height or depth.

4. Can you transform a vector field between any two coordinate systems?

Yes, it is possible to transform a vector field between any two coordinate systems. However, the equations and methods used will vary depending on the specific coordinate systems involved.

5. What are some real-world applications of transforming vector fields between cylindrical coordinates?

Transforming vector fields between cylindrical coordinates has many practical applications, such as in fluid dynamics, electromagnetism, and robotics. It is also used in fields such as engineering, physics, and astronomy to analyze and model various physical phenomena.

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