lecture notes 13

.Example. Let S be the surface (with outer unit normal vector n) of theregion D bounded by the planes z = 0, y = 0, y = 2 and the paraboliccylinder ∫∫ z = 1 − x 2 . Apply the divergence theorem to computeF · n dS, where F(x, y, z) = (x + cos y)i + (y + sin z)j + (z + e x )k.. SSolution. For any point (x, y, z) in D, we have 0 ≤ z ≤ 1 − x 2 , hence1 − x 2 ≥ 0, so x 2 ≤ 1, i.e. −1 ≤ x ≤ 1, hence we haveD = { (x, y, z) | − 1 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 1 − x 2 }.∇ · F(x, y, z) = (x + cos y) x + (y + sin z) y + (z + e x ) z = 1 + 1 + 1 = 3.Instead of evaluating the ∫∫ surface integral ∫∫∫ directly, we can∫∫∫apply thedivergence theorem that F · n dS = ∇ · F dV = 3 dV =SDD∫ 1 ∫ 2 ∫ 1−x 2∫ 13dz dy dx = 3 × 2 (1 − x 2 ) dx = 6(2 − 2 3 ) = 8.−100−1Remark. Though we had not determined the outer normal vector fieldn, but it is necessary to know that n is pointing outward on theboundary S of the solid region D, before we apply divergencetheorem.. . . . . .