Let's solve the problems step by step.

(a) Parametrize the surface $y = x^2 + \sin(z)$ where $x^2 + z^2 \leq 4$.

Step 1: Analyze the surface equation.

The equation $y = x^2 + \sin(z)$ describes a surface in 3D space with the coordinates $(x, y, z)$. To parametrize this surface, we can use the variables $x$ and $z$ as parameters since $y$ is already expressed in terms of $x$ and $z$.

The condition $x^2 + z^2 \leq 4$ defines a disk in the $xz$-plane with radius 2. Therefore, the parameters $x$ and $z$ are restricted to lie within this disk.

Step 2: Parametrize the surface.

• The parameter domain is the disk $x^2 + z^2 \leq 4$.
• For parametrization, we can use polar coordinates to cover this disk. Let $x = r \cos(\theta)$ and $z = r \sin(\theta)$, where $r \in [0, 2]$ and $\theta \in [0, 2\pi]$.

So, the parametrization is: $\mathbf{r}(r, \theta) = (r \cos(\theta), r^2 \cos^2(\theta) + \sin(r \sin(\theta)), r \sin(\theta)),$ where $r$ and $\theta$ vary as described.

Step 3: Picture the parameter domain.

The parameter domain is the disk in the $xz$-plane, with radius 2. This is a circular region centered at the origin.

(b) Parametrize the surface defined by $x^2 + y^2 - z^2 - 3 = 0$, where $-2 \leq z \leq 5$.

Step 1: Analyze the surface equation.

This equation defines a hyperboloid of one sheet. The standard equation for a hyperboloid of one sheet is: $x^2 + y^2 - z^2 = 3.$ We are asked to parametrize the part of the hyperboloid where $-2 \leq z \leq 5$.

Step 2: Parametrize the surface.

• First, rearrange the equation: $x^2 + y^2 = z^2 + 3.$
• This is a circle in the $xy$-plane with radius $\sqrt{z^2 + 3}$ for a given $z$.
• We can again use polar coordinates to parametrize the circle in the $xy$-plane.

Let: $x = \sqrt{z^2 + 3} \cos(\theta), \quad y = \sqrt{z^2 + 3} \sin(\theta),$ where $\theta \in [0, 2\pi]$ and $z \in [-2, 5]$.

Thus, the parametrization of the hyperboloid is: $\mathbf{r}(z, \theta) = (\sqrt{z^2 + 3} \cos(\theta), \sqrt{z^2 + 3} \sin(\theta), z),$ where $z \in [-2, 5]$ and $\theta \in [0, 2\pi]$.

Step 3: Picture the parameter domain.

For the parameter domain, $z$ varies between -2 and 5, and for each $z$, $\theta$ sweeps around the circle. So, the parameter domain is a rectangle in the $z\theta$-plane, with $z \in [-2, 5]$ and $\theta \in [0, 2\pi]$.

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Let me know if you need further details on any of these steps or have any questions. Also, here are some related questions you might find interesting:

1. How can we compute the surface area of a parametrized surface?
2. What are some other examples of surfaces that can be parametrized using polar coordinates?
3. How does the parametrization change for different forms of hyperboloids (e.g., two-sheeted hyperboloids)?
4. How do you interpret the domain of the parameter space geometrically?
5. What is the significance of the condition $x^2 + z^2 \leq 4$ in the first problem?

Tip: Polar coordinates are often useful for parametrizing surfaces that have circular symmetries. When you see a condition like $x^2 + z^2 \leq \text{constant}$, it’s a good clue that polar coordinates could simplify the parametrization.