Question #237591

Consider the surface S = (x, y, z) ∈ R 3 | z = 3 − x 2 − y 2 ; z ≥ 2 . Assume that S is oriented upward and let C be the oriented boundary of S. (a) Sketch the surface S in R 3 . Also show the oriented curve C and the XY-projection of the surface S on your sketch. (2) (b) Let F (x, y, z) = (2y, 3z, 4y). Evaluate the flux integral Z Z S (curl F) · n dS by i. determining curl F and the upward unit normal n of S and using the formula (17.2) on p. 104 of Guide 3 (5) ii. Using Stokes’ Theorem, convert the given flux integral to a line integral.

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