Question

Let the surface (S) be the part of the elliptic paraboloid z = x2 + 4y2lying 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 = x2 + 4y2 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 = x2 +4y2 oriented upward.

a) Parametrize the curve C and use the parametrization to evaluate the line integral

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F· d⃗r,C

where F(x, y, z) = 〈y, −xz, xz2〉.

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 x2 + y2 = 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.

Answer 1