Chapter 16: Integrals and Vector Fields - Section 16.5 - Surfaces and Area - Exercises 16.5 - Page 989: 13

(a) $r(u,v)=\lt u \cos v, u \sin v, 1-u \cos v-u \sin v\gt$ and $0 \le u \le 3$; $0 \le v \le 2\pi$ (b) $r(u,v)=\lt 1-u \cos v-u \sin v, u \cos v, u \sin v\gt$ where $0 \le u \le 3$; $0 \le v \le 2\pi$

Work Step by Step

(a) Apply polar coordinates in the $xz$ plane. We know that $r(r, \theta)=xi+yj+zk$ or, $r^2=x^2+y^2+z^2$ We have $x^2+z^2=4$ and $z=1-(x+y)$ Thus, $x=r \cos \theta, y=r \sin \theta, z=z$ Hence: $r(u,v)=\lt u \cos v, u \sin v, 1-u \cos v-u \sin v\gt$ where $0 \le u \le 3$; $0 \le v \le 2\pi$ (b) Apply polar coordinates in the $xz$ plane. We know that $r(r, \theta)=xi+yj+zk$ or, $r^2=x^2+y^2+z^2$ We have $y^2+z^2=9$ and $x=1-(y+z)$ Thus, $x=r \cos \theta, y=r \sin \theta, z=z$ Hence: $r(u,v)=\lt 1-u \cos v-u \sin v, u \cos v, u \sin v\gt$ where $0 \le u \le 3$; $0 \le v \le 2\pi$

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