Final answer:
To calculate the flux integral over the ellipsoid Σ, we can use the divergence theorem. We first calculate the divergence of the vector field F. The divergence of F is 10. Next, we calculate the volume integral of the divergence over the volume enclosed by the ellipsoid.
Step-by-step explanation:
To calculate the flux integral over the ellipsoid Σ, we can use the divergence theorem. First, we need to calculate the divergence of the vector field F. The divergence of F is given by div(F) = 2 + 5 + 3, which simplifies to div(F) = 10. Next, we calculate the volume integral of the divergence over the volume enclosed by the ellipsoid. Since the ellipsoid has the equation x²/4 + 9y² + z² = 1, we can rewrite it as x²/4 + (3y)² + (z/√1) = 1. This gives us the equation of an ellipsoid in standard form, where a = 2, b = 3, and c = 1. Using the formula for the volume of an ellipsoid, V = 4/3 * π * a * b * c, we can calculate the volume integral:
∭ V div(F) dV = 10 * (∭ V dV) = 10 * (4/3 * π * a * b * c)
Substituting the values of a, b, and c, we get:
∭ V div(F) dV = 10 * (4/3 * π * 2 * 3 * 1) = 80π