Parametric Surfaces — Question 9

Question 9 – Solution

Strategy. Each edge of the rectangular parameter domain maps to one boundary curve.

Step 1: Constant-$u$ edges $$u=1:\ \langle\cos v,\sin v,1\rangle, \qquad u=2:\ \langle 2\cos v,2\sin v,4\rangle, \quad 0\le v\le\pi.$$ These are semicircles at heights $1$ and $4$.

Step 2: Constant-$v$ edges $$v=0:\ \langle u,0,u^2\rangle, \qquad v=\pi:\ \langle-u,0,u^2\rangle, \quad 1\le u\le 2.$$ These are parabolic side curves in the plane $y=0$.

Step 3: Identify and join Since $x^2+y^2=u^2$ and $z=u^2$, the patch lies on $z=x^2+y^2$ above the half-annulus $1\le x^2+y^2\le 4$, $y\ge 0$.

Verification The four corner images are $(1,0,1)$, $(-1,0,1)$, $(2,0,4)$, and $(-2,0,4)$; each occurs as an endpoint of exactly two adjacent boundary curves.