Coordinates on the surface of a sphere of radius are the colatitude and the longitude

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Conditions of Validity, Ch A coordinate system is a map: the metric, the scale factors and where a chart fails

Problem

Coordinates on the surface of a sphere of radius

R

are the colatitude

\theta

and the longitude

\phi

. Someone computes the area element as follows. "The scale factors are

h_\theta = R

and

h_\phi = R\sin\theta

. These are the metric coefficients, so

g_{\theta\theta}=R

g_{\phi\phi}=R\sin\theta

, and the area element is

\mathrm{d}A=\sqrt{g_{\theta\theta}g_{\phi\phi}}\,\mathrm{d}\theta\,\mathrm{d}\phi = R\sqrt{\sin\theta}\,\mathrm{d}\theta\,\mathrm{d}\phi.

" ((a)) Explain the error, using a dimensional argument as well as the definition of

g_{ij}

. ((b)) Write the correct metric coefficients and area element. ((c)) Use the correct area element to find the area of the polar cap

0 \le \theta \le \pi/3

on a sphere of radius

R = 2.0\ \mathrm{m}

Browse the problem library