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UTD EE 3300 Page

EXAMINATION II

Show your work! No credit otherwise!!

  1. [15] Find the work done by the force F = exi + e-yj + ezk along the path from (0,0,0) to (1,1,1) along the curve parameterized by x=t, y=t2, z=t (0 ≤ t ≤ 1).
  1. [15] Given the force F = 2yi+5xj, find the work ∫CF•dr done by F along the closed loopthat followsthe circle (x-1)2 + (y+1) 2 = 25 in the positive direction.

3. [15] Evaluate ∫∫V•ndσ over the closed surface of the cylinder x2 + y2≤9 with 0 ≤ z ≤2 for V = cos y i + sin x j + cos z k.

  1. [15] Find the work ∫CF•dr done by F= (x2 + y2)i+(x2 + z2)j+ykalong the curve C defined by x2+y2=4 from (2,0) to (-2,0) counter-clockwise as seen from above; then along y=0 back to (2,0), all in the x-y-plane.

5. [15] Evaluate ∫C (x2 – y2)dx + (2y – x)dy, where C consists of the boundary of the region in the first quadrant that is bounded by the graphs of y = x2 and y = x3.

6. [15] Verify [that is, do it both ways] the Divergence Theorem if F = xi + yj + (z-1)k in the region bounded by the hemisphere x2 + y2 + (z-1) 2 = 9, 1 ≤ z ≤ 4, and the plane z = 1.

7. [15] Find the value of over the curved surface σ of the hemisphere given by x2 + y2 + z2 = 1, z ≥ 0 if F = yi + zj + xk. Note:

19 October 2005 “Advanced” Engineering Mathematics