Spherical Coordinates — Question 10

Question 10 – Solution

Strategy Examine the conversion formulas at the origin and on the polar axis.

Origin If $\rho=0$, then $x=y=z=0$ for every $\theta$ and $\phi$. Thus both angles are arbitrary at the origin.

Polar axis At $\phi=0$ or $\phi=\pi$, $\sin\phi=0$, so $x=y=0$ regardless of $\theta$. Therefore azimuth is arbitrary at the north and south polar rays.

Elsewhere For $\rho>0$ and $0<\phi<\pi$, the height uniquely determines $\phi=\arccos(z/\rho)$, and the nonzero projection $(x,y)$ uniquely determines $\theta$ in $[0,2\pi)$. Thus coordinates are unique away from the origin and $z$-axis.