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

[bomba923]

Just curious, why is \phi 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, \theta is calculated from the +x axis to the +y axis (counterclockwise) for position vectors.

*So, why not extrapolate alphabetically, to have the \phi be the angle between the +y axis and the position vector of a point as projected onto the yz plane?

 

[arildno]

Use whatever convention you like, there is no general agreement on this issue.

 

[bomba923]

Use whatever convention you like, there is no general agreement on this issue.

But nowhere have I found \phi calculated as the angle between the +y axis and the position vector of a point, projected onto the yz plane...

 

[amcavoy]

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).