SKKU-Calculus-Sec- 15-4 Green’s Theorem in Plane, SGLee+김태현

15.4    Green’s Theorem in the Plane, SGLee+김태현

1-4. Use Green’s Theorem to evaluate the line integral , when  is

1. , where  is a circle of , with positive orientation.

2. , where  is the positively oriented curve which is the boundary of the region bounded by .

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3. , where  is a circle of , with positive orientation.

.

let $x=rcos(t), y=rsin(t)$

4.  where  is a boundary of the rectangle , with positive orientation.

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5-8. Verify Green's Theorem by evaluating the line integral  (a) directly, (b) using Green's Theorem, where  is:

5.  with  is a closed curve of the region bounded by  and , with positive orientation.

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6.  with  is a boundary of the region defined by  and , with positive orientation.

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7.  with  is a boundary of the region enclosed by the circles  and

, with positive orientation.

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8. , where  is a circle of , with positive orientation.

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9.  , with positive orientation.

Let . To evaluate the line integral, we use Green's Theorem as follows

.

10.  where  is the arc of the parabola  in the plane  from  to .

Using Green's Theorem

11. Calculate the area bounded by the ellipse . Hence deduce the area bounded by the circle

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(a)

.

,

,

Put , area of circle : .

12. Find the area under arc of the cycloid .

13. Find the area under of one arch of the asteroid .

14. Find the area of the loop of the folium of Descartes .

15. Let  be smooth functions satisfying the following differential equations:

. Evaluate the line integral ,

where  is traversed in the positive direction of the curve .

Using Green's Theorem, we have

.

16. Let  be a solution of a system of differential equations
, where  are smooth functions in variables .

Assume further that  is periodic with a periodicity ,

namely   for all . Suppose that the solution curve (phase trajectory) is

a simple closed curve in the phase plane (plane) and  is the region bounded by .

Prove that .

Proof :  Recalling Green's Theorem,

.

Now parametrizing by , the line integral becomes

.

Therefore,  we obtain.

17. Evaluate the integral , where  is the boundary of the triangular region

with corners  with positive orientation.

18. Let  be the straight lines from  to , from  to , from  to , and from  to . Evaluate .

Let . Then

19. Let  and  is an any closed curve containing the origin. Find .

Let’s consider the region  which is the annulus as shown in the figure. Let  and  denote the outer and the inner circles of .

Then by Green’s Theorem, we have

.

Using , we can compute

Thus the vector field  is not conservative. Note that curl. Here  is not a simply connected domain.

The 6th Asian Math Conference (AMC 2013) at Busan