Chapter 17, Problem 68RE

Stokes’ Theorem on a compound surface Consider the surface S consisting of the quarter-sphere x² + y² + z² = a², for z ≥ 0 and x ≥ 0, and the half-disk in the yz-plane y² + z² ≤ a², for z ≥ 0. The boundary of S in the xy-plane is C, which consists of the semicircle x² + y² = a², for x ≥ 0, and the line segment [–a, a] on the y-axis, with a counterclockwise orientation. Let F = 〈2z – y, x – z, y – 2x〉.

a.    Describe the direction in which the normal vectors point on S.

b.    Evaluate ${\displaystyle {\oint }_{C}F\cdot dr}$.

c.    Evaluate ${\displaystyle {\iint }_S\left(\nabla \times \text{F}\right)\cdot ndS}$ and check for agreement with part (b).