To convert the given equation in rectangular coordinates $z = x^2 + y^2 - 1$ to cylindrical coordinates, let’s proceed with the following steps:

Step 1: Recall the Conversion Formulas

In cylindrical coordinates:

• $x = r \cos \theta$
• $y = r \sin \theta$
• $r^2 = x^2 + y^2$

Step 2: Substitute $x^2 + y^2$ with $r^2$

The given equation is: $z = x^2 + y^2 - 1$

Using the relationship $x^2 + y^2 = r^2$, we substitute to get: $z = r^2 - 1$

Final Answer

The equation in cylindrical coordinates is: $z = r^2 - 1$

Would you like further details or have any questions on this?

Here are 5 related questions for deeper understanding:

1. How do you convert equations from rectangular to spherical coordinates?
2. What is the geometric interpretation of $z = r^2 - 1$ in cylindrical coordinates?
3. How would this equation look if converted to spherical coordinates?
4. How can we visualize this surface in three-dimensional space?
5. What would the equation look like if a constant was added to the original rectangular equation?

Tip: When converting equations to cylindrical coordinates, always start by identifying terms that can be replaced by $r$ and $z$ based on known formulas.