1. ## parametrization

The parametrization of the sphere:

$\displaystyle {\bf x}\,(u^1,u^2) = (r \cos u^2 \cos u^1,\ r \cos u^2 \sin u^1,\ r \sin u^2)$
$\displaystyle 0 \leq u^1 < 2\pi,\ \ \ -\frac{\pi}2 \leq u^2 \leq \frac{\pi}2$

appears in my book without explanation. Any help on how to derive it? (I played with it and found the coordinates correspond to latitude and longitude).

2. What do you know about "spherical coordinates"? In standard spherical coordinates, we have $\displaystyle \rho$, the straight line distance from the origin to the point, $\displaystyle \theta$, the angle a line from the origin to the point on the xy-plane directly below the given point makes with the x-axis (the same angle as in two-dimensional polar coordinates, and $\phi$ the line from the origin to the point makes with the xy-plane. You can think of the two angles as being "longitude" and "co-latitude" (since it is measured from the "north pole" rather than the "equator").

In spherical coordinates, $\displaystyle x= \rho cos(\theta)sin(\phi)$, $\displaystyle y= \rho sin(\theta) sin(\phi)$, and $\displaystyle z= \rho cos(\phi)$.

Here, I notice we have "sin" in z rather than "cos". Okay, they are using "latitude" rather than "co-latitude". That is, they are using $\displaystyle \psi= \pi/2- \phi$ so that
$\displaystyle x= \rho cos(\theta)cos(\psi)$
$\displaystyle y= \rho sin(\theta)cos(\psi)$
$\displaystyle z= \rho sin(\psi)$

Finally, note the changes in notation: r instead of $\displaystyle \rho$, $\displaystyle u ^1$ instead of $\displaystyle \theta$, and $\displaystyle u^2$ instead of $\displaystyle \psi$.

3. Originally Posted by HallsofIvy
In spherical coordinates, $\displaystyle x= \rho cos(\theta)sin(\phi)$, $\displaystyle y= \rho sin(\theta) sin(\phi)$, and $\displaystyle z= \rho cos(\phi)$.
How did you get this? Thanks.