Let's tackle both parts one by one.

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

The surface is given by the equation $y = x^2 + \sin(z)$, and we are told that the domain of the surface is constrained by $x^2 + z^2 \leq 4$. This condition suggests that the domain is a disk in the $xz$-plane.

To parametrize the surface:

• Let $x = r \cos(\theta)$ and $z = r \sin(\theta)$, where $r$ and $\theta$ are polar coordinates. This parametrization ensures that $x^2 + z^2 = r^2$, where $r$ is constrained by $r^2 \leq 4$ (i.e., $r \leq 2$).

• From the surface equation $y = x^2 + \sin(z)$, we can express $y$ in terms of $r$ and $\theta$.

Thus, the parametrization becomes:

$\mathbf{r}(r, \theta) = \left( r \cos(\theta), r^2 \cos^2(\theta) + \sin(r \sin(\theta)), r \sin(\theta) \right)$ with $0 \leq r \leq 2$ and $0 \leq \theta < 2\pi$.

Picture of the parameter domain:

The domain is a disk in the $xz$-plane, where $x^2 + z^2 \leq 4$. In polar coordinates, this disk has radius 2, centered at the origin. The parameter domain consists of all $r$ and $\theta$ values within this disk.

2. Parametrize the part of the hyperboloid of one sheet $H$ defined by $x^2 + y^2 - z^2 - 3 = 0$ where $-2 \leq z \leq 5$.

The given equation of the hyperboloid is:

$x^2 + y^2 - z^2 = 3$

This is the equation for a hyperboloid of one sheet, which is symmetric with respect to the $z$-axis. To parametrize this surface:

• Use cylindrical coordinates for $x$ and $y$: $x = r \cos(\theta)$ and $y = r \sin(\theta)$.

• Substituting these into the equation for the hyperboloid:

$r^2 - z^2 = 3$

Solving for $r^2$, we get:

$r^2 = z^2 + 3$

Thus, $r = \sqrt{z^2 + 3}$. Therefore, the parametrization of the surface is:

$\mathbf{r}(z, \theta) = \left( \sqrt{z^2 + 3} \cos(\theta), \sqrt{z^2 + 3} \sin(\theta), z \right)$

where $-2 \leq z \leq 5$ and $0 \leq \theta < 2\pi$.

Picture of the parameter domain:

The parameter domain for $z$ ranges from $-2$ to $5$, and for each $z$, $\theta$ spans a full circle from $0$ to $2\pi$. So, the domain is a strip in the $z\theta$-plane with $-2 \leq z \leq 5$ and $\theta$ going around a circle.

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Would you like further details or clarification on any part? Here are some related questions to deepen your understanding:

1. How does the parametrization of a surface help in visualizing its geometry?
2. Can you describe how the surface for part 1 behaves as $z$ increases?
3. What would happen if the condition $x^2 + z^2 \leq 4$ were changed to something else?
4. How does the parametrization of the hyperboloid ensure that we cover the entire surface?
5. Could you explore the connection between hyperboloids and other quadric surfaces like ellipsoids?

Tip: For a clearer understanding, try sketching the parameter domain and comparing it with the actual surface's shape. This helps in visualizing how the parametrization maps the domain to the surface.