Sample Questions for Third Vector Analysis Exam
April 2009 1. Give an example of a vector field with curl always equal to y i or say why this is impossible. 2. Calculate the curl of the function F = y sin xz i − z cos xy j + +y cos xz k. 3. Let F = xy i − yxj, and let C be the curve in the xy plane formed by joining the part of the of the circle of radius 2 in the second quadrant with the line from (−2, 0) to (2, 0). Evaluate C F · dR. 4. Determine whether or not the following fields are conservative. For those that are not, explain why; for those that are, find a potential function. (a.) F = y 2 z 3 i + 2xyz 3 j + (3xy 2 z 2 + 2z)k, and (b.) F = (z cos xz + zey )i + (xzey + yz)j + (x cos xz + xey )k. 5. Let F = z i+yx2 z 2 j+xk, and let C be the helix γ(t) = − cos 2ti+sin 2tj+6tk from the point (−1, 0, 12π) to the point (0, 1, 27 π). Evaluate C F · dR. 2 6. Let F = y i − z j + xk, and let S be the surface of the region enclosed by the planes z = 0, y = 0, x = 0 and the plane through the points (1, 0, 0), (0, 1, 0) F · dS. and (0, 0, 1). This is an irregular tetrahedron. Evaluate S 7. Let f (x, y, z) = 2y, and let R be the region enclosed by the planes z = 0, y = 0, x = 0 and the plane through the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Evaluate R f dV .
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