Question

# Suppose that S is the portion of the surface {z = 5 − x2 − y2} conﬁned to the region {0 ≤ z ≤ 1}. Let F

= (h−2xy ,y ,3x). (a) Find the area of S. (b) Let n be the unit normal vector to S pointing in the positive z-direction. Calculate ZZS(curl F)·ndS . (c) Let E be the solid whose boundary is made of S capped oﬀ with the disk of radius 2 located at height z = 1 and with the disk of radius √5 located at height z = 0 (both disks have their center on the z-axis). Calculate ZZ∂E n·FdS , where n is the outer normal unit vector to E.