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 positivedirection of z? axis (the inner normal vector to the surface).Further, let C denotes the curve of the intersection of theparaboloid z = x2 + 4y2 and the plane z = 1 orientedcounterclockwise when viewed from positive z? axis above the planeand let S1 denotes the part of the plane z = 1 inside theparaboloid z = x2 +4y2 oriented upward.
a) Parametrize the curve C and use the parametrization toevaluate the line integral
?
F· d?r,C
where F(x, y, z) = ?y, ?xz, xz2?.
b) Find G = ? × F, where F(x,y,z) is the vector field from Parta), parameterize the surface S1 and use the parametrization toevaluate the flux of the vector field G.
HINT: The area enclosed by an ellipse x2 + y2 = 1 is ab?.
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.
