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MATH 2374 Midterm: MATH 2374 UMN Spring 08 Exam 2sol

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turquoisegnu128

31 Jan 2019

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Exam 2 solutions: (30 points) (a) (3 points each) curl( f ) yes curl(div f) no; div f is a real-valued function, and curl applies to vector elds. div(curl f) yes. 2 (( sin t)2 + (cos t)2 + 22)1/2 dt = z 4 . 5 dt = 2 5. (c) (15) since ||c (t)|| = 5 as above, we have. 2 ln 2: (20 points) since f = (yz2, xz2, 2xyz) is the gradient of the function f (x, y, z) = xyz2, by the fundamental theorem we have. F ds = f (3, 1, 2) f (1, 2, 3) = 12 ( 18) = 30. One may also parametrize the path and calculate the line integral directly. Zc: (20 points) for f = (p, q) = (x2, xy), we have qx py = y. F ds = zzr y da, where r is the triangular region bounded by c+.

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