# Test4mac2313 page 4 of 6 7 10 pts compute the work in

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TEST4/MAC2313 Page 4 of 6 ______________________________________________________________________ 7. (10 pts.) Compute the work in moving a particle along of the path in the xy-plane that goes from the point (-1,-1) to the point (1,1) along the curve y = x 5 against the force field defined by F ( x , y ) y x , yx 3 . The Toidi’s parameterization for the curve, in the correct direction, is given in a vector form by r ( t ) = < t , t 5 > for t ε [-1,1]. Consequently, r ( t ) = < 1 , 5 t 4 > and thus, w C F d r C F ( r ( t )) r ( t ) dt 1 1 ( t 5 t )(1) ( t 5 t 3 )(5 t 4 ) dt 1 1 t 5 t 5 t 12 dt 10 1 0 t 12 dt 10 13 . How did Em Toidi know that a parameterization is needed?? First, the field is NOT CONSERVATIVE. [Go check this, Frodo.] This means that the Fundamental Theorem of Line Integrals CANNOT BE USED. Second, the curve is NOT CLOSED, although the curve is simple. As a consequence, Green’s Theorem CANNOT BE USED. This means finally, you are stuck with the "definition." A nice, easy parameterization is a must. This is actually an easy path integral, oddly. Or was it evenly????? [E.T. used both!!] ______________________________________________________________________ 8. (10 pts.) Starting at the point (0,0), a particle goes along the y-axis until it reaches the point (0,4). It then goes from (0,4) to (-4,0) along the circle with equation x 2 + y 2 = 16. Finally the particle returns to the origin by travelling along the x-axis. Use Green’s Theorem to compute the work done on the particle by the force field defined by F ( x , y ) = < -5 y 3 , 5 x 3 > for ( x , y ) ε 2 . [Draw a picture. This is easy??] W C F d r C 5 y 3 dx 5 x 3 dy R x (5 x 3 ) y ( 5 y 3 ) dA 15 R x 2 y 2 dA 15 π π /2 4 0 r 3 drd θ 480 π . [ Corrected .] Picture: Quarter disk of radius 4 in the second quadrant bounded by the y- axis on the east and the x-axis on the south. Trace the boundary counter- clockwise.

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TEST4/MAC2313 Page 5 of 6 ______________________________________________________________________ 9. (10 pts.) (a) Show that the vector field F ( x , y ) < cos( x ) e y
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