The exercise appears to be focused on identifying equations that describe surfaces in Cartesian, cylindrical, and spherical coordinates. The goal is to select three equations that describe a surface representing a sphere or spherical shell.

Here is a breakdown of the exercise:

Key points from the text:

1. Task: Identify equations that describe a surface that is a sphere or spherical shell.
2. Hints:
   • Convert the given equations to Cartesian coordinates to make comparisons easier.
   • Focus on the $xy$-plane (where $y = 0$) for easier comparison.

Provided equations:

1. $\rho = 2 \sin(\phi)$ in spherical coordinates.
2. $x^2 + y^2 + z^2 - 2x - 2y = -2$ in Cartesian coordinates.
3. $x^2 + y^2 - 2x = 1$ in Cartesian coordinates.
4. $x^2 + y^2 - 2x - 2y = 0$ in Cartesian coordinates.
5. $\sqrt{r^2 + z^2} = 2$ in cylindrical coordinates.
6. $r = 2 \cos(\phi)$ in spherical coordinates.
7. $r = 2 \cos(\phi)$ in cylindrical coordinates.

Process for solving:

• Convert all equations to Cartesian form (where necessary) and check if they match the equation of a sphere: $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = R^2$ which is the general form of a sphere in Cartesian coordinates.

Equations to investigate:

• Equation 1: $\rho = 2 \sin(\phi)$ (spherical coordinates).
• Equation 2: $x^2 + y^2 + z^2 - 2x - 2y = -2$ (Cartesian coordinates).
• Equation 5: $\sqrt{r^2 + z^2} = 2$ (cylindrical coordinates).

These three likely describe spherical surfaces.

Would you like a detailed breakdown of each equation conversion to confirm the solution?

Here are five questions for further exploration:

1. How do you convert equations from spherical to Cartesian coordinates?
2. What is the geometric interpretation of cylindrical coordinates in relation to Cartesian?
3. Can you derive the equation of a sphere in cylindrical coordinates?
4. How does the equation $r = 2 \cos(\phi)$ describe a surface in cylindrical coordinates?
5. Why do we focus on simplifying the equation for the $xy$-plane?

Tip: Remember that converting between coordinate systems often involves using trigonometric identities and definitions like $x = r \cos(\theta)$ and $z = \rho \cos(\phi)$.