Why Is \(\phi\) Defined Differently in Spherical Coordinates?

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Discussion Overview

The discussion revolves around the definition of the angle \(\phi\) in spherical coordinates, specifically why it is defined as the angle between the +z axis and the position vector of a point projected onto the yz plane. Participants explore the conventions used in spherical coordinates compared to those in polar and cylindrical coordinates.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the convention of defining \(\phi\) as the angle from the +z axis, suggesting it might be more intuitive to define it as the angle from the +y axis instead.
  • Another participant asserts that there is no general agreement on the convention used for \(\phi\), indicating variability in definitions.
  • A third participant notes that they have not encountered the definition of \(\phi\) as the angle from the +y axis in any sources.
  • A later reply mentions that understanding the convention becomes clearer when working on problems involving intersections of geometric shapes, such as spheres and cones, and suggests that using spherical coordinates simplifies certain calculations.

Areas of Agreement / Disagreement

Participants express differing views on the definition of \(\phi\) in spherical coordinates, indicating that multiple competing views remain without a consensus on the best convention.

Contextual Notes

The discussion highlights the lack of a universally accepted convention for defining \(\phi\) and the potential implications for problem-solving in spherical coordinates.

bomba923
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Just curious, why is [tex]\phi[/tex] calculated as the angle between the +z axis and a position vector of a point of a function, as projected onto the yz plane? Why this convention?

In polar & cylindrical, [tex]\theta[/tex] is calculated from the +x axis to the +y axis (counterclockwise) for position vectors.

*So, why not extrapolate alphabetically, to have the [tex]\phi[/tex] be the angle between the +y axis and the position vector of a point as projected onto the yz plane? :smile:
 
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Use whatever convention you like, there is no general agreement on this issue.
 
arildno said:
Use whatever convention you like, there is no general agreement on this issue.

But nowhere have I found [tex]\phi[/tex] calculated as the angle between the +y axis and the position vector of a point, projected onto the yz plane...
 
You will see why once you start working with problems involving these. For example, take the intersection between a sphere and a cone (I posted about this before). This can be computed by a double integral, but if you use a triple integral and spherical coordinates, it becomes much simpler (thanks to HallsofIvy on this).
 

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