MAT 2322 Final: MAT 2322 University of Ottawa Final Exam exam 3+Solutions
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Question 1. a) div ~f = yz + 2 x2yz2 + 3 y2z2: since this is not zero, divergence test tells us that ~f cannot be the curl of any vector eld. Solve grad f = (2xy 2x)~i + (x2 2y)~j. to get the critical points: 2xy 2x = 0, x2 2y = 0, (x, y) = (0, 0) and ( . At (0,0): fxx = 2y 2, fxy = 2x, fyy = 2. Since fxx(0, 0) = 2 < 0, (0,0) is a relative maximum. D = 0( 2) (2 2)2 = 8 < 0. D = 0( 2) ( 2 2)2 = 8 < 0. In spherical coordinates, the same region can be described as: 4 sin d d d = . This problem is very easy to solve if you notice that ~f is the gradient of f = 2x2 + 3y2. Then, by the fundamental theorem of calculus for line integrals, we get:
