To construct a vector-valued function $\mathbf{r}(u, v)$ whose graph represents the given surface, let’s break it down step-by-step.

Problem Recap:

1. Surface: $z = x^2 + y^2$ (a paraboloid).
2. Constraint: Inside the cylinder $x^2 + y^2 = 9$.
3. $u$ and $v$ will be the parameters of the function.

Solution:

Parameterization of the Surface

1. Use cylindrical coordinates where: $x = r\cos\theta, \quad y = r\sin\theta, \quad z = r^2$ Here, $r^2 = x^2 + y^2$, and $r$ must satisfy $0 \leq r \leq 3$ because the cylinder is bounded by $x^2 + y^2 = 9$ (which corresponds to $r = 3$).

2. Assign the parameters $u = r$ and $v = \theta$, where:

   • $u$ represents the radial distance ($0 \leq u \leq 3$).
   • $v$ represents the angle ($0 \leq v \leq 2\pi$).

3. The vector-valued function is: $\mathbf{r}(u, v) = \langle u\cos v, u\sin v, u^2 \rangle$

Final Answer:

$\mathbf{r}(u, v) = \langle u\cos v, u\sin v, u^2 \rangle, \quad 0 \leq u \leq 3, \quad 0 \leq v \leq 2\pi$

Would you like me to explain any part of this in more detail or verify it with examples?

Related Questions:

1. What is the geometric significance of the parameterization in cylindrical coordinates?
2. How would you find the surface area of the portion of the paraboloid defined here?
3. What changes would occur if the cylinder's radius were different (e.g., $x^2 + y^2 = 16$)?
4. How would you describe the normal vector to the surface at a point $\mathbf{r}(u, v)$?
5. How would you convert this parameterization to Cartesian form for verification?

Tip:

When parameterizing a surface bounded by a cylinder or similar constraint, always identify the region of interest in cylindrical or spherical coordinates for easier parameterization.