Understanding Spherical Coordinates and Their Range

[madachi]

Homework Statement

I am confused about spherical coordinates stuff. For example, we can parametrize a sphere of radius 3 by

$x = 3 sin \phi cos \theta$
$y = 3 sin \phi sin \theta$
$z = 3cos\phi$

where $0 \le \theta \le 2 \pi$ and $0 \le \phi \le \pi$ .

I don't understand about the range of $\phi$.

1) Why is $0 \le \phi \le \pi$ ?
2) If we only want the lower hemisphere, why is the range now $\frac{\pi}{2} \le \phi \le \pi$ ?
3) What about the range of $\phi$ if we want the upper hemisphere?

Is there any place where I get to see the diagram so I can get the picture better?

Thanks!

 

[Pengwuino]

1) I assume you mean why doesn't $$\phi$$ go to $$2\pi$$? By allowing $$\theta$$ to go to $$2\pi$$, you cover the entire sphere. Any point in the space that you feel you could get by extending $$\phi$$ to $$2\pi$$ is covered by simply moving to $$\theta > \pi$$.

2) Look at how the coordinate system is defined and notice that you must go beyond $$\frac{\pi}{2}$$ to be in the lower hemisphere. As for 3), same idea, just look at a diagram as to how the coordinate system is defined to see why certain ranges are why they are.

http://upload.wikimedia.org/wikiped...nates.svg/429px-Spherical_Coordinates.svg.png

This is a diagram of how spherical coordinates are typically defined. NOTE: Your definition of the coordinates have $$\theta$$ and $$\phi$$ switched.

 

[phyzguy]

It's just like latitude and longitude on the Earth. Longitude (normally called phi, but what you called theta) runs from 0 to 360 degrees (2 pi), but latitude (normally called theta but what you called phi) only needs to run from -90 degrees to +90 degrees (total range of 180 degrees, or pi) to cover the sphere. The only difference is that on the Earth we define the equator as 0 degrees, the north pole as +90 degrees and the south pole as -90 degrees, whereas in physics, we usually define the north pole as 0 degrees, the equator as 90 degrees (pi/2), and the south pole as 180 degrees (pi).