Parametric Surfaces — Question 1

Question 1 – Solution

Strategy. Read $x=u$ and $y=v$, eliminate the parameters, and map the rectangle’s four corners.

Step 1: Identify the surface Since $x=u$ and $y=v$, $$z=1+2x-y.$$ Thus the image is the portion of this plane above the rectangle $0\le x\le 2$, $0\le y\le 1$.

Step 2: Tangent and normal vectors $$\mathbf r_u=\langle 1,0,2\rangle, \qquad \mathbf r_v=\langle 0,1,-1\rangle.$$ Therefore $$\boxed{\mathbf r_u\times\mathbf r_v=\langle-2,1,1\rangle}.$$

Step 3: Map the corners $$\begin{align*} \mathbf r(0,0)&=(0,0,1),& \mathbf r(2,0)&=(2,0,5),\\ \mathbf r(0,1)&=(0,1,0),& \mathbf r(2,1)&=(2,1,4). \end{align*}$$ These four points bound the parallelogram patch.

Verification The normal $\langle-2,1,1\rangle$ is perpendicular to both tangent vectors, and each corner satisfies $z=1+2x-y$.