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 ) isnt 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 , so that F ( r ( t )) = sin t , cos t and r ( t ) = sin t , cos t . Then C F d r = 2 F ( r ( t )) r ( t ) dt = 2 1 dt = 2 so the field isnt conservative. What justifies the last step? ..?.. Theorem 2: the line integral of a conservative vector field around a closed loop equals 0. Now that Ive shown you that the field isnt conservative, Im going to show you that it IS conservative! Let be the angle coordinate of ( x , y ) in polar coordinates. = arctan( y / x ), so x = 1/(1+( y / x ) 2 ) ( y / x 2 ) = y /( x 2 + y 2 ) = P and y = 1/(1+( y / x ) 2 ) (1/ x ) = 1/(1+( y / x ) 2 ) ( x / x 2 ) = x /( x 2 + y 2 ) = Q . So = F . Whats 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, is not equal to arctan( y / x ) outside the 1st and 4th quadrants. ) outside the 1st and 4th quadrants....
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11.18 - Section 13.3: The fundamental theorem for line...

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