Write the equations in cylindrical coordinates.

(a)
3z = 2x^2 + 2y^2

(b)
7x^2 + 7y^2 = 4y

(a) Sure, here you go:

In cylindrical coordinates, the equation 3z = 2x^2 + 2y^2 becomes:
3z = 2ρ^2 (since x = ρcosθ and y = ρsinθ)

(b) Now let's tackle the equation 7x^2 + 7y^2 = 4y:

In cylindrical coordinates, the equation 7x^2 + 7y^2 = 4y becomes:
7(ρcosθ)^2 + 7(ρsinθ)^2 = 4ρsinθ

Simplifying it further, we have:
7ρ^2(cos^2θ + sin^2θ) = 4ρsinθ
7ρ^2 = 4ρsinθ

Hope that brings a smile to your face!

To write the equations in cylindrical coordinates, we need to express the variables x, y, and z in terms of the cylindrical coordinates ρ, φ, and z.

(a) 3z = 2x^2 + 2y^2

To express x and y in terms of cylindrical coordinates, we use the following conversions:
x = ρ*cos(φ)
y = ρ*sin(φ)
z = z

Substituting these equations into the given equation:

3z = 2(ρ*cos(φ))^2 + 2(ρ*sin(φ))^2
3z = 2ρ^2*cos^2(φ) + 2ρ^2*sin^2(φ)

Simplifying further:

3z = 2ρ^2*(cos^2(φ) + sin^2(φ))
3z = 2ρ^2

The equation in cylindrical coordinates is 3z = 2ρ^2.

(b) 7x^2 + 7y^2 = 4y

Using the cylindrical coordinate conversions, we have:
x = ρ*cos(φ)
y = ρ*sin(φ)
z = z

Substituting these values into the given equation:

7(ρ*cos(φ))^2 + 7(ρ*sin(φ))^2 = 4(ρ*sin(φ))

Simplifying further:

7ρ^2*cos^2(φ) + 7ρ^2*sin^2(φ) = 4ρ*sin(φ)

7ρ^2*(cos^2(φ) + sin^2(φ)) = 4ρ*sin(φ)

Since cos^2(φ) + sin^2(φ) = 1, we can simplify further:

7ρ^2 = 4ρ*sin(φ)

Dividing both sides by ρ:

7ρ = 4*sin(φ)

The equation in cylindrical coordinates is 7ρ = 4*sin(φ).

To express the equations in cylindrical coordinates, we need to replace the Cartesian variables (x, y, z) with their equivalent cylindrical variables (ρ, φ, z).

(a) 3z = 2x^2 + 2y^2

In cylindrical coordinates, x = ρcos(φ), y = ρsin(φ), and z = z. Substituting these values into the given equation, we get:

3z = 2(ρcos(φ))^2 + 2(ρsin(φ))^2

Simplifying this equation further:

3z = 2ρ^2cos^2(φ) + 2ρ^2sin^2(φ)

Using the trigonometric identity cos^2(φ) + sin^2(φ) = 1:

3z = 2ρ^2(1)

Therefore, the equation in cylindrical coordinates is: 3z = 2ρ^2.

(b) 7x^2 + 7y^2 = 4y

Similarly, substituting x = ρcos(φ) and y = ρsin(φ) into the equation, we have:

7(ρcos(φ))^2 + 7(ρsin(φ))^2 = 4(ρsin(φ))

Simplifying:

7ρ^2cos^2(φ) + 7ρ^2sin^2(φ) = 4ρsin(φ)

Using the trigonometric identity cos^2(φ) + sin^2(φ) = 1:

7ρ^2(1) = 4ρsin(φ)

Therefore, the equation in cylindrical coordinates is: 7ρ^2 = 4ρsin(φ).

(a) z = 2/3 r^2

(b)
7r^2 = 4r sinθ
r = 4/7 sinθ