GreensTh

GreensTh - Department of Mathematics University of Southern California Greens Theorem Examples Sec 13-4 Chapter-End Review Examples Page 800

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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 . Thepart ic leismovedfrom A (3 , 0 , 0) to B (0 , 1 2 π , 3) along two di f erent paths: (a) along a straight line (b) along the helix: x =3co s t, y = t, z =3s in 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 f ne r ( t ). (a) For the straight line from A (3 , 0 , 0) to B (0 , 1 2 π , 3), we may de f 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 f ne r ( t )=( 3 c o s t ) ˆ i + t ˆ j
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This note was uploaded on 09/07/2011 for the course MATH 39578 taught by Professor Penner during the Spring '09 term at USC.

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GreensTh - Department of Mathematics University of Southern California Greens Theorem Examples Sec 13-4 Chapter-End Review Examples Page 800

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