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GreensTh - Department of Mathematics University of Southern...

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Department of Mathematics University of Southern California April 15, 2011 Green’s Theorem Examples Sec 13-4, Chapter-End Review Examples, Page 800 Problem 10 The force on a particle is given by F ( x, y, z ) = z ˆ i + x ˆ j + y ˆ k . The particle is moved from A (3 , 0 , 0) to B (0 , 1 2 π , 3) along two di ff erent paths: (a) along a straight line (b) along the helix: x = 3 cos t, y = t, z = 3 sin t . Calculate the work done in each case. Solution W = 8 C F · d r = 8 C F · r I ( t ) dt. In order to carry out this integral, we need to de fi ne r ( t ). (a) For the straight line from A (3 , 0 , 0) to B (0 , 1 2 π , 3), we may de fi ne r ( t ) = < 3 , 0 , 0 > + t < 3 , 1 2 π , 3 > = (3 3 t ) ˆ i + 1 2 π t ˆ j + 3 t ˆ k r I ( t ) = 3 ˆ i + 1 2 π ˆ j + 3 ˆ k F ( t ) = 3 t ˆ i + (3 3 t ) ˆ j + 1 2 π t ˆ k The parameter t lies in the range, 0 t 1. Therefore, W = 8 1 0 F ( t ) · r I ( t ) dt = 8 1 0 ± 3 t ˆ i + (3 3 t ) ˆ j + 1 2 π t ˆ k = · ± 3 ˆ i + 1 2 π ˆ j + 3 ˆ k = dt = 8 1 0 ± 9 t + (3 3 t ) 1 2 π + 3 2 π t = dt = ± 9 2 t 2 + (3 t 3 2 t 2 ) 1 2 π + 3 4 π t 2 = 1 0 = ± 9 2 + (3 3 2 ) 1 2 π + 3 4 π = = ± 9 2 + 3 2 π = (b) For the helix from A (3 , 0 , 0) to B (0 , 1 2 π , 3), we may de fi ne r ( t ) = (3 cos t ) ˆ i + t ˆ j + 3(sin t ) ˆ k , 0 t 1 2 π r I ( t ) = ( 3 sin t ) ˆ i + ˆ j + 3(cos t ) ˆ k F ( t ) = (3 sin t ) ˆ i + (3 cos t ) ˆ j + t
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