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# 15_2_9 - F is conservative on 3 except on the plane z = 0...

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9. F = 2 x z i + 2 y z j - x 2 + y 2 z 2 k , F 1 = 2 x z , F 2 = 2 y z , F 3 = - x 2 + y 2 z 2 . We have F 1 y = 0 = F 2 x , F 1 z = - 2 x z 2 = F 3 x , F 2 z = - 2 y z 2 = F 3 y . Therefore, F may be conservative in 3 except on the plane z = 0 where it is not defined. If F = φ , then ∂φ x = 2 x z , ∂φ y = 2 y z , ∂φ z = - x 2 + y 2 z 2 . Therefore, φ( x , y , z ) = Z 2 x z dx = x 2 z + C 1 ( y , z ) 2 y z = ∂φ y = C 1 y C 1 ( y , z ) = y 2 z + C 2 ( z ) φ( x , y , z ) = x 2 + y 2 z + C 2 ( z ) - x 2 + y 2 z 2 = ∂φ z = - x 2 + y 2 z 2 + C 0 2 ( z ) C 2 ( z ) = 0 . Thus φ( x , y , z ) = x 2 + y 2 z is a potential for
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Unformatted text preview: F is conservative on 3 except on the plane z = 0. The equipotential surfaces have equations x 2 + y 2 z = C , or Cz = x 2 + y 2 . Thus the equipotential surfaces are circular paraboloids....
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