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13 Nov 2019
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
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
Elin HesselLv2
11 Apr 2019