1. **Problem:** Consider the surface in $\mathbb{R}^3$ parameterized by $\vec{\Phi}(r,\theta) = (r \cos \theta, r \sin \theta, \theta)$ with $0 \leq r \leq 1$ and $0 \leq \theta \leq 4\pi$. (a) Sketch and describe the surface. 2. **Step 1: Understand the parameterization** - The parameter $r$ controls the radius in the $xy$-plane, from 0 to 1. - The parameter $\theta$ controls the angle around the $z$-axis and also the height $z=\theta$. 3. **Step 2: Describe the surface shape** - For each fixed $\theta$, the point lies on a circle of radius $r$ in the $xy$-plane at height $z=\theta$. - As $\theta$ increases from 0 to $4\pi$, the height increases linearly. - The radius $r$ varies from 0 to 1, so the surface is a spiral-shaped surface extending upward from $z=0$ to $z=4\pi$. 4. **Step 3: Visualize the surface** - The surface looks like a spiral ramp or a helicoid with radius expanding from the center (axis) out to 1. - At $r=0$, the surface is the $z$-axis line from $z=0$ to $z=4\pi$. - At $r=1$, the surface traces a helix of radius 1 around the $z$-axis. 5. **Summary:** The surface is a spiral or helicoid-like surface formed by circles of radius $r$ stacked along the $z$-axis with height $z=\theta$ and angle $\theta$ wrapping around the axis from 0 to $4\pi$. Final answer: The surface is a helicoid-like spiral surface extending from $z=0$ to $z=4\pi$ with radius $r$ from 0 to 1, wrapping twice around the $z$-axis.