3. Let the surface (S) be the part of the elliptic paraboloid z = x^2 + 4y^2 lying below the plane z = 1. We define the orientation of (S) by taking the unit normal vector n pointing in the positive direction of z- axis (the inner normal vector to the surface). Further, let C denotes the curve of the intersection of the paraboloid z = x^2 + 4y^2 and the plane z = 1 oriented counterclockwise when viewed from positive z- axis above the plane and let S1 denotes the part of the plane z = 1 inside the paraboloid z = x^2 + 4y^2 oriented upward.
a) Parametrize the curve C and use the parametrization to evaluate the line integral
∠ F ∙ dr,
where F(x, y, z) = <y, -xz, xz^2>.
b) Find G = ∇ × F, where F(x, y, z) is the vector field from Part a), parameterize the surface S1 and use the parametrization to evaluate the flux of the vector field G.
HINT: The area enclosed by an ellipse x^2/a^2 + y^2/b^2 = 1 is abπ.
c) What is the flux of the vector field G = ∇ × F, from Part b), across the surface (S)? Explain why the answers in a), b), and c) must be the same.