Final answer:
The equation z = x² + y² - 6 in cylindrical coordinates is z = r² - 6 by substituting x with r*cos(θ) and y with r*sin(θ) into the original equation.
Step-by-step explanation:
Converting Rectangular to Cylindrical Coordinates
To find the equation of the surface z = x² + y² - 6 in cylindrical coordinates, we recall that cylindrical coordinates are related to rectangular coordinates by the relationships: x = r cos(θ), y = r sin(θ), and z = z. Here, r is the radial distance from the origin, θ is the azimuth angle, and z represents the same vertical placement as in rectangular coordinates.
Plugging the expressions for x and y in terms of r and θ into the given equation:
Replace x with r cos(θ) and y with r sin(θ).
Thus, we get z = (r cos(θ))² + (r sin(θ))² - 6.
Simplify the expression knowing that cos(θ)² + sin(θ)² = 1, which gives us z = r² - 6.
The cylindrical equation for the surface represented by the given rectangular equation z = x² + y² - 6 is z = r² - 6.