## Thomas' Calculus 13th Edition

$r(u,v)=\lt \sqrt 3 \sin u \cos v,\sqrt 3 \sin u \sin v, \sqrt 3 \cos u \gt$ and $\dfrac{\pi}{3} \le u \le \dfrac{2\pi}{3};\\0 \le v \le 2 \pi$
Use spherical coordinates as: $x= l \sin \phi \cos \theta; y= l \sin \phi \sin \theta; z= l \cos \phi$ ; $0 \le \phi \le \pi; \\0 \le \theta \le 2 \pi$. We know that $r(r, \theta)=xi+yj+zk$ or, $r^2=x^2+y^2+z^2$ We have: $x^2+y^2+z^2 =3$ $x= \sqrt 3 \sin u \cos v, y= \sqrt 3 \sin u \sin v, z= \sqrt 3 \cos u$ So, $r(u,v)=\lt \sqrt 3 \sin u \cos v,\sqrt 3 \sin u \sin v, \sqrt 3 \cos u \gt$ and $\dfrac{\pi}{3} \le u \le \dfrac{2\pi}{3};\\0 \le v \le 2 \pi$