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13 Nov 2019
l f F. Tds, where F = (ys_y,-2x3), over all simple closed curves (oriented counterclockwise). (Use Green's theo- (2) Fi nd the maximum value of the integra rem, and consider where the integrand is non-negative.)
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Keith Leannon
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10 Jun 2019
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Related questions
#19, #21, #24
Divergence theorem Problem 19. Let V be the solid bounded by the paraboloid 8-a2 -3y2 from above and by the saddle surface2 from below. Find the flux of outward through the boundary of V Problem 20. Let V be the solid bounded by the paraboloid z = 6-x2-y2 from above and by the cone z = VE2+32 from below. Find the flux of F = (z, z, y) outward through the boundary of V. Stokes's theorem Problem 21. Find the circulation of F (y, zy, ) along the curve oriented counterclockwise as seen from above. Problem 22. Find the circulation of F (2yz - ry, 2rz, ry -y) along the curve oriented counterclockwise as seen from above. e Green's formula Problem 23. Use Green's formula to prove that the area of a region bounded by a simple closed curve C is equal to Use this formula to calculate the area of the elliptic diskS1 Problem 24. Calculate the integral where C is the curve ë + ly-1 oriented Counterclockwise
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
5. Consider the following integral, which does not seem very easy to evaluate. 2Ï eloo co s2(t) sin(1 + e30cow(t) )sin(t) + cos2(t) dt In this problem we will evaluate the integral by using Green's theorem. Let c be the circle of radius 1 centered at (0,0), and oriented counterclockwise. One possible param- eterization of c is c(t) (cos(t), sin(t)) with t ⬠[0, 2 (a) Find a vector field F so that when evaluating , F ds using the parameterization above, the integral that results is () (b) Use Green's theorem to convert this to an integral over the unit disc, and evaluate that integral.
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