Solutions8

Solutions8 - 1 Mathematics 1c: Solutions, Homework Set 8...

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Unformatted text preview: 1 Mathematics 1c: Solutions, Homework Set 8 Due: Tuesday, June 1 at 10am. 1. (10 Points) Section 8.1, Exercises 3c and 3d . Verify Greens theorem for the disk D with center (0 , 0) and radius R and P ( x,y ) = xy = Q ( x,y ) and the same disk for P = 2 y,Q = x. . Solution. For 3c, let c ( t ) = ( R cos t,R sin t ) be the parameterization of D . Then Z D P dx + Qdy = Z 2 ( R 2 cos t sin t,R 2 cos t sin t ) (- R sin t,R cos t ) dt =- R 3 Z 2 sin 2 t cos tdt + R 3 Z 2 cos 2 t sin tdt = 0 + 0 = 0 . Also, ZZ D Q x- P y dxdy = ZZ D ( y- x ) dxdy = Z R Z 2 ( r sin - r cos ) r d dr = Z R (0 + 0) r 2 dr = 0 . Hence, Greens theorem for 3c is verified. For 3d, note that Greens theorem Z D Pdx + Qdy = ZZ D Q x- P y dxdy becomes Z D 2 y dx + xdy = ZZ D (1- 2) dxdy =- ZZ D dxdy The right side is- R 2 while the left side is, since x = R cos and y = R sin , Z 2 (2 R sin )(- R sin ) d + ( R cos )( R cos ) d =- 2 R 2 Z 2 sin 2 d + R 2 Z 2 cos 2 d. Using the fact that sin 2 and cos 2 have averages 1 2 , namely 1 2 Z 2 sin 2 d = 1 2 (this is one way of remembering the formula for the integrals of sin 2 and cos 2 on [0 , 2 ] and [0 ,...
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Solutions8 - 1 Mathematics 1c: Solutions, Homework Set 8...

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