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13 Nov 2019
14. (4 points) Let F(x,y) =Farctan (y) = x2+92i+x2 If C is the unit circle parameterized by r(t) = costi + sin tj, 0 A. 0 B. T C. 2Ï D. Undefined E. None of the above IS 27, then F·dr =
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Keith Leannon
Lv2
10 Oct 2019
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Related questions
ven by the vector equations r(s, )u,ss,Acmu the a printe) identily the surface gives A plane B elliptic paraboloid C ylinder D ellipsoid E None of the above a differentiable function of r and y with continuous second order partial derivatives. Let Mi + Nj be a gradient field and let Cbe the (positively oriented) ellipse ns shown in the sketch 17. (4 points) Let f be F-â½js below. Consider the following statements. (a) F is a conservative vector field. (o) (curl F) k> A. All three statements are true. B. Only (a) and (b) are true. C. Only (a) and (e) are true. D. Only (b) and (c) are true. E. None of the above ) = V arctan It C is the unit circle parameterized by r A. t) = costi + sin tj. 0
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(1 point) Let be the radial force field F = x + y . Find the work done by this force along the following two curves, both which go from (0, 0 to (1, 1) (Compare your answers!) A. If G is the parabola: x = t, y = t2, 0 1 1 , then B. If C2 is the straight line segment: x = 112, y = 1t2, 0 F·dr = C1 F·dr = C2 t 1, then
EXAMPLE 5 If F(x, y) = (-yi + xj)/(x2 + y2), show that fF . d. 2Ï for every positively oriented simple closed path that encloses the origin SOLUTION Since C is an arbitrary closed path that encloses the origin, it's difficult to compute the given integral directly. So let's consider a counterclockwise-oriented circle C' with center the origin and radius n, where n is chosen to be small enough that C" lies inside C. (See the figure.) Let D be the region bounded by C and C'. Then its positively oriented boundary is C U (-C1) and so the general version of Green's Theorem gives dA (x2 +y) (x2 +y?)2 We now easily compute this last integral using the parametrization given by r(t) = ncos(t)i + nsin(t)j, 0 Thus, t 2Ï 2Ï (-nsin(t))(-nsin(t))+(1-ncos(t) |X )(ncos(t)) dt n2(cos(t))2 n(sin(t))2
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