0.0.0.12 6.

$\begin{displaymath} \frac{\rd\Vector{\varphi}}{\rd u}(u,v)= \left( \begin{array... ...ay}{c} -R\cos u\sin v R\cos u\cos v 0 \end{array}\right) \end{displaymath}$

$\displaystyle \frac{\rd\Vector{\varphi}}{\rd u}\times \frac{\rd\Vector{\varphi}}{\rd v}$	$\displaystyle =\det \begin{pmatrix}-R\sin u\cos v & -R\cos u\sin v & \Vector{e}... ...sin v & R\cos u\cos v & \Vector{e}_2 R\cos u & 0 & \Vector{e}_3 \end{pmatrix}$
	$\displaystyle = \begin{pmatrix}-R^2\cos^2u\cos v -R^2\cos^2u\sin v -R^2\s... ...matrix}-R^2\cos^2u\cos v -R^2\cos^2u\sin v -R^2\sin u\cos u \end{pmatrix}$
	$\displaystyle =-R^2\cos u \begin{pmatrix}\cos u\cos v \cos u\sin v \sin u \end{pmatrix}.$

$\displaystyle \left\Vert \frac{\rd\Vector{\varphi}}{\rd u}\times \frac{\rd\Vector{\varphi}}{\rd v} \right\Vert =R^2\cos u.$

$\displaystyle \mu_c(S)=\dint_{-\pi/2\le\theta\le\pi/2\atop -\pi\le\phi\le\pi} R... ...}^{\pi/2}\cos u\;\D u\int_{-\pi}^\pi \D v =R^2\cdot 2\cdot 2\pi=4\pi R^2. \qed$