Evaluate the surface integral
S
F · dS
for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = x i + y j + z4 k
S is the part of the cone
z = sqrt(x2 + y2)
beneath the plane
z = 3
with upward orientation
To evaluate the surface integral of a vector field F across a surface S, you can use the formula:
∫∫S F · dS
where F is the vector field and dS is the differential of surface area on S.
In this case, the given vector field is F(x, y, z) = x i + y j + z^4 k.
The surface S is described as the part of the cone z = √(x^2 + y^2) beneath the plane z = 3, with the upward orientation.
To evaluate the surface integral, we need to parameterize the surface S. We can use cylindrical coordinates (ρ, θ, z), where ρ represents the distance from the origin to a point in the xy-plane and θ is the angle made with the positive x-axis.
First, let's find the limits of integration for each variable:
- ρ varies from 0 to 3 (since z = 3 is the upper boundary)
- θ varies from 0 to 2π to cover the entire circle
- z varies from 0 to √(x^2 + y^2)
Now, let's express the surface S in terms of the parameterization:
x = ρ cos(θ)
y = ρ sin(θ)
z = √(x^2 + y^2)
Using these parameterizations, we can compute the necessary derivatives to find the differential of surface area dS:
∂r/∂ρ = cos(θ) i + sin(θ) j
∂r/∂θ = -ρ sin(θ) i + ρ cos(θ) j
∂r/∂z = (∂z/∂x) i + (∂z/∂y) j = (ρ/√(x^2 + y^2)) (x i + y j)
Now, we can calculate the dot product F · dS:
F · dS = (x i + y j + z^4 k) · (∂r/∂ρ cross ∂r/∂θ) dρ dθ
= (x i + y j + z^4 k) · (cos(θ) i + sin(θ) j) cross (-ρ sin(θ) i + ρ cos(θ) j) dρ dθ
= (x i + y j + z^4 k) · (ρ cos(θ) j - ρ sin(θ) i) dρ dθ
= ρx cos(θ) - ρy sin(θ) + z^4 ρ dρ dθ
= (ρ cos(θ) ) (ρ) dρ dθ
= ρ^2 dρ dθ
Now, the flux across S is given by the surface integral:
∫∫S F · dS = ∫∫S ρ^2 dρ dθ
Integrating over the limits of ρ (0 to 3) and θ (0 to 2π), we get:
∫0 to 2π ∫0 to 3 ρ^2 dρ dθ
Evaluating the integral gives the final result of the surface integral for the given vector field and surface.