## Thomas' Calculus 13th Edition

$r(u,v)=2 \cos v \space i+u \space j+2 \sin v \space k;\\ -2 \le u \le 2;\space \\ 0 \le v \le \pi$
Apply polar coordinates in $xz$ plane. We know that $r(r, \theta)=xi+yj+zk$ or, $r^2=x^2+y^2+z^2$ We have $x^2+z^2=4$ Now, $r(x,z)=2 \cos \theta \space i+uj+2 \sin \theta \space k$ Consider $y=u; \theta=v$ This yields: $r(u,v)=2 \cos v \space i+u \space j+2 \sin v \space k$; $-2 \le u \le 2$ and $0 \le v \le \pi$ Hence: $r(u,v)=2 \cos v \space i+u \space j+2 \sin v \space k;\\ -2 \le u \le 2;\space \\ 0 \le v \le \pi$