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hw5 - Mathematics 20E Spring 2009 Homework 5 Please...

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Mathematics 20E - Spring 2009 - Homework 5 May 26, 2009 Please complete the following problems and turn them in to the homework drop box in the sixth floor of AP&M labelled “Math 20E / Ryan Szypowski” by Friday, June 5 at 5:30pm. Only a subset of the problems will be graded; leave some blank at your own risk. 1. In this problem, we will show that a particular vector field is path independant from first principles without invoking any theorems about conservative vector fields. Consider the constant vector field F = i + j + k . Let C 1 and C 2 be any two oriented curves with the same starting point and ending point. That is, let c 1 : [ a 1 ,b 1 ] 3 and c 2 : [ a 2 ,b 2 ] 3 be parameterizations of C 1 and C 2 respectively. Then, knowing only that c 1 ( a 1 ) = c 2 ( a 2 ) and c 1 ( b 1 ) = c 2 ( b 2 ), show integraldisplay C 1 F · d s = integraldisplay C 2 F · d s . 2. Recall that if F ( x,y,z ) = F 1 ( z,y,z ) i + F 2 ( x,y,z ) j + F 3 ( x,y,z ) k satisfies ∇· F = 0 then F = ∇× G for some G . Prove that the following 1
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construction of G works: G ( x,y,z ) = G 1 ( x,y,z ) i + G 2 (
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