11.18 - Section 13.3 The fundamental theorem for line...

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Unformatted text preview: Section 13.3: The fundamental theorem for line integrals (continued) Discuss Group Work 1 Why F = P i + Q j with P ( x , y ) = – y /( x 2 + y 2 ) and Q ( x , y ) = x / ( x 2 + y 2 ) isn’t conservative on its domain R 2 \ {(0,0)}: Let C be the unit circle, traversed once in the counterclockwise direction from (1,0) to (1,0). Put r ( t ) = 〈 x ( t ), y ( t ) 〉 = 〈 cos t , sin t 〉 with 0 ≤ t ≤ 2 π , so that F ( r ( t )) = 〈 –sin t , cos t 〉 and r ′ ( t ) = 〈 – sin t , cos t 〉 . Then ∫ C F • d r = ∫ 2 π F ( r ( t )) • r ′ ( t ) dt = ∫ 2 π 1 dt = 2 π ≠ so the field isn’t conservative. What justifies the last step? ..?.. Theorem 2: the line integral of a conservative vector field around a closed loop equals 0. Now that I’ve shown you that the field isn’t conservative, I’m going to show you that it IS conservative! Let θ be the angle coordinate of ( x , y ) in polar coordinates. θ = arctan( y / x ), so θ x = 1/(1+( y / x ) 2 ) (– y / x 2 ) = – y /( x 2 + y 2 ) = P and θ y = 1/(1+( y / x ) 2 ) (1/ x ) = 1/(1+( y / x ) 2 ) ( x / x 2 ) = x /( x 2 + y 2 ) = Q . So ∇θ = F . What’s going on? (It was suggested in class that the problem is that at x = 0, the subexpression y / x blows up. This is a symptom of the fact that, contrary to what I claimed, θ is not equal to arctan( y / x ) outside the 1st and 4th quadrants. ) outside the 1st and 4th quadrants....
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This note was uploaded on 02/13/2012 for the course MATH 241 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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11.18 - Section 13.3 The fundamental theorem for line...

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