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Transforming a vector in spherical coordinates

  1. Apr 1, 2014 #1
    I've got a Green's function in which all the impulses are on the line from the north pole to the origin (polar angle θ=0) and terminating with a point impulse at the north pole. I've found its gradient at a field point, and I want to rotate everything to a new coordinate system with the source line at arbitrary θ and azimuthal angle Ω. What I've done is:
    1. Find the angle ψ between the field point and the source.
    2. Use ψ in the formula which returns the gradient for a source on the line to the north pole. Note that since the basis vectors use the north pole as a reference location, this effectively has the source's location as a reference location.
    3. Since the result of step 2 is the field in terms of the basis vectors the source's location, I next transform the basis vectors to a coordinate system in which the source is not necessarily on the line through the north pole.

    For the angle between the field point and and the source I found:
    ψ = arccos(sin(θfield)*sin(θsource)*cos(Ωfieldsource)+cos(θfield)*cos(θsource))

    The r-component is unchanged. Calling the vector B, and with ψ being the polar coordinate in its old source-based coordinate system, the θ component becomes:

    However, when I tested it for θsource=0 (so it should reproduce the original result) I had a different answer than before. I had trouble finding a reference. Can you help me find my error, or help me find a reference which uses the same method I can compare this to?
  2. jcsd
  3. May 4, 2014 #2
    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
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