MATH V1207 Final: MATH V1205 Columbia Fall01Final
31 Jan 2019
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Write your name and my name on each exam booklet. You can earn partial credit only if you justify your steps. Start each problem on a new page: (10 points) solve the di erential equation y + = 4: (10 points) let x > 0. Zc sin2(z) dz (a) 2 i (b) 2 (c) 2/3 (d) sin3(2 i)/3 (e) 0: (10 points) let c be the circle of radius 2 , centered at the origin. 0 z 1 x2+y2 f (x, y, z) dz dy dx: (10 points) suppose that f is a vector eld in 3-space everywhere perpendicular to a surface s with boundary c. show that. Z zs ( f ) ds = 0: (10 points) suppose that div(f ) > 0 inside the unit ball, x2 + y2 + z2 1. Show that f cannot be everywhere tangent to the surface of the sphere.
