Calculate the circulation, SF dr, in two ways, directly and using Stokes' Theorem. The vector field F = 8yi - 8xj and C is the boundary of S, the part of the surface z = 16x² - y² above the xy-plane, oriented upward. Note that C' is a circle in the xy-plane. Find a 7(t) that parameterizes this curve. r(t) +5 with (Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) With this parameterization, the circulation integral is Sc F.dr = dt, where a and b are the endpoints you gave above. Evaluate your integral to find the circulation: SF. dr = Using Stokes' Theorem, we equate f. dr = Noting that the surface is given by z = 16 - x² - y², find dà = dy dx. curl F. dÃ. Find curl F = With R giving the region in the xy-plane enclosed by the surface, this gives Ss curl F. dà = SR dy dx. Evaluate this integral to find the circulation: ScF. dr = f curl F. dà =

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