15_2_4 - ln ( x 2 + y 2 ) is a scalar potential for F , and...

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4. F = x i + y j x 2 + y 2 , F 1 = x x 2 + y 2 , F 2 = y x 2 + y 2 . We have F 1 y = - 2 xy ( x 2 + y 2 ) 2 = F 2 x . Therefore, F may be conservative. If F = φ , then ∂φ x = x x 2 + y 2 , ∂φ y = y x 2 + y 2 . Therefore, φ( x , y ) = Z x x 2 + y 2 dx = ln ( x 2 + y 2 ) 2 + C 1 ( y ) y x 2 + y 2 = ∂φ y = y x 2 + y 2 + c 0 1 ( y ) c 0 1 ( y ) = 0 . Thus we can choose C 1 ( y ) = 0, and φ( x , y ) = 1 2
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Unformatted text preview: ln ( x 2 + y 2 ) is a scalar potential for F , and F is conservative everywhere on 2 except at the origin....
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