Chapter 17.7, Problem 41E

Compound surface and boundary Begin with the paraboloid z = x² + y², for 0 ≤ z ≤ 4, and slice it with the plane y = 0. Let S be the surface that remains for y ≥ 0 (including the planar surface in the xz–plane) (see figure). Let C be the semicircle and line segment that bound the cap of .S in the plane z = 4 with counterclockwise orientation. Let F = 〈2z + y, 2x + z, 2y + x〉.

a.    Describe the direction of the vectors normal to the surface that are consistent with the orientation of C.

b.    Evaluate ${\displaystyle {\iint }_S\left(\nabla \times F\right)\cdot ndS}$

c.    Evaluate ${\displaystyle {\oint }_{C}F\cdot dr}$and check for agreement with part (b).