Write the equations in cylindrical coordinates.
(a)
3z = 2x^2 + 2y^2
(b)
7x^2 + 7y^2 = 4y
To write the equations in cylindrical coordinates, we need to convert the variables from Cartesian coordinates (x, y, z) to cylindrical coordinates (ρ, θ, z). Here's how you can do it:
(a) 3z = 2x^2 + 2y^2
First, we need to replace x and y with their respective cylindrical coordinates.
In cylindrical coordinates, x = ρ cos(θ) and y = ρ sin(θ). So we have:
3z = 2(ρ cos(θ))^2 + 2(ρ sin(θ))^2
Simplifying this expression, we get:
3z = 2ρ^2cos^2(θ) + 2ρ^2sin^2(θ)
Since cos^2(θ) + sin^2(θ) = 1, we can simplify further:
3z = 2ρ^2(1)
Therefore, the equation in cylindrical coordinates is:
3z = 2ρ^2
(b) 7x^2 + 7y^2 = 4y
Similarly, we need to replace x and y with their respective cylindrical coordinates.
In cylindrical coordinates, x = ρ cos(θ) and y = ρ sin(θ). So we have:
7(ρ cos(θ))^2 + 7(ρ sin(θ))^2 = 4(ρ sin(θ))
Simplifying this expression, we get:
7ρ^2cos^2(θ) + 7ρ^2sin^2(θ) = 4ρsin(θ)
Again, using the identity cos^2(θ) + sin^2(θ) = 1, we can simplify further:
7ρ^2(1) = 4ρsin(θ) - 7ρ^2sin^2(θ)
Therefore, the equation in cylindrical coordinates is:
7ρ^2 = 4ρsin(θ) - 7ρ^2sin^2(θ)
These are the equations in cylindrical coordinates for the given expressions.