Why Are the Limits Different for Phi and Theta in Spherical Coordinates?

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Discussion Overview

The discussion revolves around the limits of integration for spherical coordinates, specifically why the limits for the angle phi (φ) are from 0 to 2π while those for theta (θ) are from 0 to π. Participants explore the implications of these limits in the context of volume integrals and geometric interpretations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of different limits for φ and θ, suggesting that both could potentially range from 0 to π.
  • Another participant explains that integrating φ from 0 to 2π allows for symmetric coverage of points in the spherical coordinate system, while θ ranges from 0 to π to cover the full vertical span from the positive to negative z-axis.
  • A different perspective is introduced using a metaphor involving a string fixed at the North Pole, illustrating how sweeping the string around covers different regions of the globe.
  • One participant notes the analogy of latitude and longitude, indicating that latitude ranges from -90 to 90 degrees and longitude from -180 to 180 degrees, which parallels the limits of θ and φ.
  • There is a humorous acknowledgment of the confusion caused by physicists swapping the definitions of θ and φ in different contexts.
  • Another participant shifts the focus to a related topic about understanding integration limits for various shapes, indicating a desire for resources to clarify these concepts.
  • A metaphorical example is provided about programming a robot to paint a sphere, emphasizing the importance of correctly defining limits to ensure complete coverage.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and implications of the limits for φ and θ, indicating that multiple competing interpretations exist without a clear consensus on the matter.

Contextual Notes

Some participants express uncertainty about their mathematical understanding, which may influence their interpretations of the limits in spherical coordinates.

Davio
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Hey guys, could one of you explain why when doing a volume integral using spherical polar coordinates, you have the limits as 2 pi to 0 on phi but only pi to 0 on theta? Thanks.

To clarify, I've been doing this all this time for questions, but it just occurred to me that I Don't know why i do that :-p.
 
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Hi Davio! :smile:

(Of course, you can't go from o to 2π on both, because then you'd be covering every point twice!)

Well, you can always do it the other way round … but the integrals that usually occur in practice just happen to be symmetric in phi, so integrating from 0 to 2π on phi is dead easy! :biggrin:

In particular, if you're converting from (x,y,z), then you get ∫∫∫(r^2)sinthetadrdthetadphi … and that itself is symmetric in phi, so with luck the whole thing immediately becomes 2π∫∫r^2)sinthetadrdtheta. :smile:
 
Hmmm, but wouldn't it be, for both phi and theta, just pi to zero - ie. half each? Or am imagining this wrong :-p. R is radius, so integrating along that place, in a circle, theta and phi are both angles, just in different directions? My maths is a bit poor (not good for a physics major)!
 
Hi, Davio!

Take a line segment from the centre of the ball to its surface. That line segment makes the angle theta with the "z"-axis.

In order to reach ALL points on the circle with the same angle to the "xy"-plane, you rotate the line segment around the z-axis, with phi then going from 0 to 2pi.

Now, you will have covered ALL these circles (and hence all points) for theta-values going from 0 (i.e, the line segment runs along the positive z-half axis) to pi, (i.e along the negative half axis for z)
 
… don't ignore America …

Yup, arildno is right!

Start with a very long string, fixed at the North pole. Take the other end down the Greenwich meridian from theta = 0 to π. Now you're at the South pole.

So far so good … :smile:

Now sweep the string round from phi = 0 to π. You'll cover most of Europe and Africa and the whole of Asia and Australia.

But you'll stop at the International Date Line!

:rolleyes: What about America? :rolleyes:​
 
One simple way to see it. Latitude goes from -90 to 90, while longitude goes from -180 to 180.
 
It's those blasted physicists again! They keep swapping \theta and \phi on us!
 
I'm going to sit down and think about the replies in a minute, I'm on to a question about conical cones now, does anyone have any good resources for understand the images behind integration? I can integrate etc, but can't quite understand the limits of weird shapes, or even normal shapes!
 
… painting problem …

Hi Davio! :smile:

Do you mean comical cones?

Now, they are weird! :biggrin:

But keep this in perspective … this isn't an integration problem … it's only a painting problem.

Imagine you have to program a robot to paint a sphere … do you tell it to paint from 0 to π, or 2π, for each coordinate?

If you give it the wrong instructions, it'll either waste paint or not use enough! :frown:

That's your only problem … making sure that everything is covered which should be! :smile:
 

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