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14.9 The Divergence Theorem Let E be a simple solid region with the boundary surface S (which is a closed surface.) Let S be positively oriented (i.e.the orientation on S is outward that is, the unit normal vector n is directed outward from E ). . Let E be a simple solid region whose boundary surface S has positive (outward) orientation. Let F be a continuous vector eld on an open region that contains E . Then ¨ ˚ Divergence Theorem F · dS = S div F dV E Example 1. Let E = { (x, y, z) | x2 + y 2 ≤ 16, 0 ≤ z ≤ 3 }. Find the ux of the vector eld F = h x3 , 2xez , 3y 2 z i over ∂E . Example 2. Evaluate ˜ S curl F·dS if S is the boundary of the ellipsoid and F is an arbitrary continuous vector eld. x2 y 2 z 2 + + ≤1 4 9 16 Example 3. Let F = h x3 , 2xz 2 , 3y 2 z i and E be the solid bounded by the paraboloid z = 4 − x2 − y 2 and the xy -plane. (a) Find ˜ ∂E F · dS. (b) Find ˜ S1 F · dS where S1 is the part of the paraboloid z = 4 − x2 − y 2 between the planes z = 0 and z = 4.