To convert the given equation $x^2 + y^2 + z^2 - 4z = 0$ into cylindrical coordinates, we use the following relationships between rectangular and cylindrical coordinates:

1. $x = r \cos\theta$
2. $y = r \sin\theta$
3. $x^2 + y^2 = r^2$
4. $z = z$ (remains unchanged in cylindrical coordinates).

Substituting into the equation:

The given equation is: $x^2 + y^2 + z^2 - 4z = 0$ Substitute $x^2 + y^2 = r^2$: $r^2 + z^2 - 4z = 0$

Simplify further:

Combine terms involving $z$: $r^2 + (z^2 - 4z) = 0$ Complete the square for the $z$-terms: $z^2 - 4z = (z - 2)^2 - 4$ Substitute this back into the equation: $r^2 + (z - 2)^2 - 4 = 0$ Simplify: $r^2 + (z - 2)^2 = 4$

Final equation in cylindrical coordinates:

The equation in cylindrical coordinates is: $r^2 + (z - 2)^2 = 4$

This represents a cylinder centered along the axis $z = 2$ with a radius of 2.

Do you want further clarification or details? Here are some related questions:

1. What is the geometric interpretation of the final equation in cylindrical coordinates?
2. How do you derive cylindrical coordinates transformations in general?
3. Can you convert this equation into spherical coordinates as well?
4. How does the equation change if the cylinder is rotated in space?
5. How would you graph this equation in a 3D coordinate system?

Tip: When completing the square, always double-check your signs and constants for accuracy.