math226fall07pfs

# math226fall07pfs - 1 . a) Given ~ F ( x,y ) = h P,Q i =- y/...

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Unformatted text preview: 1 . a) Given ~ F ( x,y ) = h P,Q i =- y/ ( x 2 + y 2 ) , x/ ( x 2 + y 2 ) , check whether Q x = P y and find R C ~ F · d~ r for C : x 2 + y 2 = 1; Answer . ∂P/∂y =- ( x 2 + y 2 )+2 y 2 ( x 2 + y 2 ) 2 = ∂Q/∂x = ( x 2 + y 2 )- 2 x 2 ( x 2 + y 2 ) 2 . Since C : ~ r ( t ) = h cos t, sin t i , ≤ t ≤ 2 π, ~ F ( ~ r ( t )) =- sin t/ (cos 2 t + sin 2 t ) , cos t/ (cos 2 t + sin 2 t ) = h- sin t, cos t i and ~ r ( t ) = h- sin t, cos t i , Z C ~ F · d~ r = Z 2 π (sin 2 t + cos 2 t ) dt = 2 π. b) Given ~ F ( x,y ) = h P,Q i = x/ ( x 2 + y 2 ) , y/ ( x 2 + y 2 ) , check whether Q x = P y and find R C ~ F · d~ r for C : x 2 + y 2 = 1 . Answer . ∂P/∂y =- 2 yx ( x 2 + y 2 ) 2 = ∂Q/∂x =- 2 xy ( x 2 + y 2 ) 2 . Since C : ~ r ( t ) = h cos t, sin t i , ≤ t ≤ 2 π, ~ F ( ~ r ( t )) = cos t/ (cos 2 t + sin 2 t ) , sin t/ (cos 2 t + sin 2 t ) = h cos t, sin t i and ~ r ( t ) = h- sin t, cos t i , Z C ~ F · d~ r = Z 2 π (- sin t cos t + cos t sin t ) dt = 0 . 2 . Let ~ F = P ~ i + Q ~ j + R ~ k = y ~ i + ( x + z ) ~ j + y ~ k. a) Is ~ F conservative? If so, find its potential function. b) Evaluate R C ~ F · d~ r along the line segment from (0 , , 0) to (1 , 1 , 3) ....
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## This note was uploaded on 12/16/2011 for the course MATH 226 at USC.

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math226fall07pfs - 1 . a) Given ~ F ( x,y ) = h P,Q i =- y/...

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