Plot vector functions in different coordinate systems?

Questions
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1

Hello to all:

I teach physics at a community college in Arizona. Currently I am working on documents to assist students in visualization of Electrostatic and Magnetic fields. Almost all require either cylindrical or spherical coordinates. My question is: Given a field expressed in cylindrical coordinates, how do you generate a vector plot of the field in that coordinate system?

Please note, that I have already found a method to convert the field from cylindrical to cartesian coordinates, and the VectorPlot function works just fine once I convert it, but I would like some options so that I do not need to convert the field but rather just plot it directly. This would also be helpful in the event that I want to visualize fields in other orthogonal coordinate systems e.g. toroidal coordinates.

Thank you all for your consideration and responses.

Chris Ubing
Sierra Vista, Az

2

Seems not to be available directly by the usage of options to VectorPlot[]. But if e.g. there are the tangential vectors to a centered circle to be plotted

VectorPlot[{-y, x}/Norm[{x, y}], {x, -3, 3}, {y, -3, 3}]

this is in cylindrical co-ordinates {0, 1} or with other words AngleVector[ \phi] or with even other words

VectorPlot[AngleVector[ArcTan[-y, x]], {x, -3,
3

You may use TransformedField to convert to Cartesian coordinates first, then use VectorPlot in the plane (or VectorPlot3D in 3-space) on the resulting field.

4

Yes,

VectorPlot[Evaluate[
    TransformedField["Polar" -> "Cartesian", {0, 1}, {r, \[Theta]} -> {x, y}]
  ], {x, -3, 3}, {y, -3, 3}]