Parametric Surfaces — Question 3

Question 3

Parametrize the half-cylinder $$x^2+y^2=4,\qquad y\ge 0,\qquad -1\le z\le 3,$$ and choose the orientation whose normal points outward.

Tasks

1. Give a parameter domain that covers the patch once, apart from boundary seams.

2. Compute a normal from the tangent vectors.

3. Verify that the normal is outward.

Question 3 – Solution

Strategy. Use an angular parameter for the circular direction and a height parameter for $z$.

Step 1: Parametrize $$\mathbf r(\theta,s)=\langle 2\cos\theta,2\sin\theta,s\rangle, \qquad 0\le\theta\le\pi,\qquad -1\le s\le 3.$$ The interval $0\le\theta\le\pi$ enforces $y\ge 0$.

Step 2: Tangent vectors $$\mathbf r_\theta=\langle-2\sin\theta,2\cos\theta,0\rangle, \qquad \mathbf r_s=\langle 0,0,1\rangle.$$ Their ordered cross product is $$\boxed{\mathbf r_\theta\times\mathbf r_s =\langle 2\cos\theta,2\sin\theta,0\rangle}.$$

Step 3: Check orientation At each point, the horizontal radial vector from the $z$-axis is $\langle x,y,0\rangle=\langle 2\cos\theta,2\sin\theta,0\rangle$, exactly the cross product. Hence it points outward.

Verification Its magnitude is $2$, never zero, so the parametrization is regular everywhere on the patch, including its boundary curves.