# C let r be the solid bounded by the cylinder x 2 y 2

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(c) Let R be the solid bounded by the cylinder x 2 + y 2 = 4, the xy –plane and the plane x + z = 6 and let F ( x, y, z ) = ( x 2 + sin z, xy + cos z, e y ) . Find the flux of F outward across the surface of R . 0 2 x 0 2 y 0 4 8 z

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MATB42H page 3 (d) Evaluate integraldisplay S ω , where ω = x dy dz + y dz dx + z dx dy and S is the piece of the paraboloid z = 9 - x 2 - y 2 lying above the xy –plane, oriented by the downward unit normal. 9. [12 points] (a) Let f ( x, y, z, u ) = xyz 2 and let ω = y 2 dx + xz dy + xyzu 2 du . Compute d ( df ω ). (b) i. Define what it means for a k –form ω to be closed . ii. Define what it means for a k –form ω to be exact . (c) Let ω = (3 y 2 z 2 + sin z ) x 2 dy dz - 2 xy 3 z 2 dz dx + 2( x cos z - y ) dx dy . Show, without calculating it, that there is a 1–form η such that = ω . 10. [8 points] Let C be the curve y 2 + z 2 = 1 lying on the plane x = 1, and let ω = z dx + ( x - z ) dy . (a) Calculate integraldisplay C ω by using a parametrization of C which appears to be in the clock- wise direction when viewed from the origin. (b) Write C = ∂S for a suitably chosen surface S and, applying Stokes’ Theorem, verify your answer in part (a).
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