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Z C S The surface S is a subset of the sphere of radius 1, and is enclosed by the smooth curve C (as shown in the figure) parametriza r(t) = sint costi + sint sintj + costk, ists 2 (a) [6pt] Note that r( 2") = (0, 1, 0), r(0) = (0, 0, 1), r( 7) = (0, 1, 0) . As such, let C = C1 UC2, where C1 connects (0, 1, 0) up to (0, 0, 1) and C2 connects (0, 0, 1) down to (0, 1, 0) (as shown in the figure). Write parameterizations ri (t) for the curve C1 and r2(t) for the curve C2. (b) [6 pt] Find the region in the 60-plane that gets mapped to S by r($, 0) = (sin ocos 0, sin psin 0, cos d). Hint. The following trig identities are helpful: COS(7 - x) = - cost, cos(7 + x) = - cost, sin(7 - x) = sinx, sin(7 +x) = - sinc. The region should look like this figure (you need to provide the details). (c) [6 pt] Compute the surface area of S. (d) [6 pt] Let T denote the unit tangent vector field to the curve C, and let F(x, y, z) = (y, -x,0). Compute the line integral o F . Tds by integrating over the curve C. (e) [6 pt] Use Stokes's Theorem to compute o F . Tds