Question

Compute the flux of the vector field F = 2x²y² - zk through the surface S, which is the cone √x² +y²= 2z, with 0 < z < R, oriented downward.

(a) Parameterize the cone using cylindrical coordinates (write 0 as theta):
x(r, theta) =
y(r, theta) =
z(r, theta) =

Answer 1

To parameterize the cone using cylindrical coordinates, we can use the equations x = rcos(θ), y = rsin(θ), and z = z. Ignoring θ = 0, the parameterization can be written as x = rcos(θ), y = rsin(θ), and z = R.

Step-by-step explanation:

To parameterize the cone using cylindrical coordinates, we can use the equations:

x = rcos(θ)

y = rsin(θ)

z = z

Given that we are ignoring θ = 0, we can write the parameterization as:

x = rcos(θ)

y = rsin(θ)

z = R

where r is the radius of the cone and θ is the angle of rotation around the z-axis.

Answer 2

The computation of flux through a cone involves parameterizing the cone in cylindrical coordinates, finding the surface element, and integrating the dot product of the vector field and the surface element over the surface.

Step-by-step explanation:

The question asks to compute the flux of the vector field F = 2x²y² - zk through a cone with certain parameters. The first step is to parameterize the cone using cylindrical coordinates. In cylindrical coordinates, the parameterization of the cone can be given by:

• x(r, θ) = r cos(θ)
• y(r, θ) = r sin(θ)
• z(r, θ) = r/2 since √x²+y²=2z leads to r=2z

Next, we need to calculate the surface element dA in cylindrical coordinates which incorporates the Jacobian for the transformation from rectangular to cylindrical coordinates, and then compute the dot product of the vector field with this surface element, oriented in the downward direction, which means the normal vector would be pointing inward toward the origin, or in the negative z direction.

The flux is then obtained by integrating this dot product over the entire surface of the cone. This step involves setting up the limits of integration for r from 0 to R and θ from 0 to 2π.

The final step is carrying out the integration, which involves trigonometric and polynomial integration techniques to solve for the total flux through the surface of the cone.