Final answer:
To evaluate A using a line integral, we need to calculate the curl of the vector field F and then take its dot product with the outward unit normal vector n. Applying the surface integral formula, we find the curl of F as (2y, -2x, -2z), and the unit normal vector as (0, 0, 1). After simplifying the dot product, we set up the integral over the upper half of the ellipsoid and evaluate it using appropriate techniques like spherical coordinates or numerical integration.
Step-by-step explanation:
To evaluate A using a line integral, we need to calculate the curl of the vector field F and then take its dot product with the outward unit normal vector n. We integrate this scalar function over the surface S of the upper half of the ellipsoid.
First, let's find the curl of F:
∇ x F = (2y, -2x, -2z)
Next, let's find the unit normal vector n:
We can write the equation of the ellipsoid as:
x^2 + y^2/8 + z^2/8 = 1
By differentiating the equation with respect to z, we get:
z/4 = 0 => z = 0
Therefore, the unit normal vector is: n = (0, 0, 1)
Now we can evaluate A using the line integral:
A = ∫(2y, -2x, -2z) · (0, 0, 1) dS
As the z component of n is zero, the dot product simplifies to: A = ∫-2z dS = -2∫z dS
Since z ≥ 0 and the surface S is the upper half of the ellipsoid, we have θ = 0 to π and φ = 0 to 2π. Therefore, the integral becomes:
A = -2∫(z dsinθdφdθ) = -2∫(z sinθ) dφdθ
Using the equation of the ellipsoid, we can write z in terms of x and y:
z = √(1 - x^2 - 8y^2)
Substituting this into the integral, we get:
A = -2∫(√(1 - x^2 - 8y^2) sinθ) dφdθ
To evaluate this integral, we need to use appropriate methods such as spherical coordinates or numerical integration.