MAT 2322 Final: MAT 2322 University of Ottawa Final Exam exam 1+Solutions
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Their intersection is x2 + y2 = 4, which is a circle of radius 2. So the region can be described in cylindrical coordinates: 0 2 , r2 z 8 r2. If you get a nonzero vector eld, then it is not a gradient eld, and vice versa. curl ~f = (1 x)ey ~k, curl ~g = ~0, curl ~h = ey ~k. Thus, ~g = ey~i + xey ~j is the only gradient vector eld. Now nd the potential: f (x, y) = z ey dx = xey + c(y), fy(x, y) = xey + c (y) = xey, Thus, f (x, y) = xey + c for any constant c is a potential function for ~g. S is a graph of the function z = f (x, y) = px2 + y2. The domain w , according to the question, is the ring 1 px2 + y2 4. Using the formula for ux through graphs, we get:
