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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 \( z = \rho \cos(\phi) \).

Under finner du en samling likninger i kartesiske, sylinder- og kulekoordinater. Kryss av de likningene som beskriver en flate i rommet som er en sfære/kuleskall. Det er 3 riktige svar.

This exercise involves identifying equations in Cartesian, cylindrical, and spherical coordinates that describe a sphere or spherical shell. Explore the steps for converting between coordinate systems and understanding how to describe surfaces in 3D geometry.

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