Chapter 17.7, Problem 43E

Maximum surface integral Let S be the paraboloid z = a(1 – x² – y²), for z ≥ 0, where a > 0 is a real number. Let F = (x – y, y + z, z – x). For what value(s) of a (if any) does ${\displaystyle {\iint }_S\left(\nabla \times F\right)\cdot ndS}$have its maximum value?