Parametric Surfaces — Question 6

Question 6 – Solution

Strategy. Interpret radius, angle, and height directly, then test the tangent cross product.

Step 1: Identify In cylindrical terms, $$\rho=u, \qquad\theta=v, \qquad z=v.$$ As the angle increases, height increases at the same rate; the image is a helicoid patch.

Step 2: Coordinate curves If $u=u_0$, then $$\langle u_0\cos v,u_0\sin v,v\rangle$$ is a helix of radius $u_0$; for $u_0=0$ it is the $z$-axis. If $v=v_0$, varying $u$ traces a radial line segment at height $z=v_0$.

Step 3: Regularity $$\mathbf r_u=\langle\cos v,\sin v,0\rangle, \qquad \mathbf r_v=\langle-u\sin v,u\cos v,1\rangle,$$ and $$\boxed{\mathbf r_u\times\mathbf r_v=\langle\sin v,-\cos v,u\rangle}.$$ Its magnitude is $\sqrt{1+u^2}>0$, even at $u=0$, so the parametrization is regular everywhere.

Verification The first two components of the cross product have squared sum $1$, preventing the normal from vanishing.