11.21 - Section 13.3 wrap-up: Last time we looked at F = Pi...

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Section 13.3 wrap-up: Last time we looked at F = P i + Q j with P ( x , y ) = – y /( x 2 + y 2 ) and Q ( x , y ) = x /( x 2 + y 2 ), and I proved that the field F isn’t conservative on R 2 \ {(0,0)} by describing a closed curve C for which C F d r is non-zero. But then I calculated that F = ∇θ where θ is the angle coordinate of ( x , y ) in polar coordinates. Doesn’t this contradict Theorem 2? ..?. . My calculation ignored the fact that “jumps” from π to – as one crosses the negative x -axis. So is not even continuous, let alone differentiable, along the negative x -axis, and the equation F = ∇θ fails there. to take its values in [0,2 ) instead of (– , ]. But this shifts the discontinuity problem without removing it: the modified F = ∇θ fails on the positive x -axis. Moral: We can move the problem around, but we can’t get rid of it!)
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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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11.21 - Section 13.3 wrap-up: Last time we looked at F = Pi...

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