Calculate the circulation, feF. 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 = 4 - x² above the ay-plane, oriented upward. Note that C is a circle in the xy-plane. Find a r(t) that parameterizes this curve. F(t) = 9. with st≤ (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 ScF.dr = Sa Evaluate your integral to find the circulation: fc F. dr = dt, where a and b are the endpoints you gave above. Using Stokes' Theorem, we equate foF dr = f curl F. dÃ. Find curl F = Noting that the surface is given by z=4 - x² - y², find A = dy dr. With R giving the region in the zy-plane enclosed by the surface, this gives Is curl F. dà = SR dy dr. Evaluate this integral to find the circulation: ScF. dr = fg curl F. dà =

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