15_2_5 - = -2 xz + y 2 + C 2 ( z ) ⇒ C 2 ( z ) = . Thus...

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5. F = ( 2 xy - z 2 ) i + ( 2 yz + x 2 ) j - ( 2 zx - y 2 ) k , F 1 = 2 xy - z 2 , F 2 = 2 yz + x 2 , F 3 = y 2 - 2 zx . We have F 1 y = 2 x = F 2 x , F 1 z = - 2 z = F 3 x , F 2 z = 2 y = F 3 y . Therefore, F may be conservative. If F = φ , then ∂φ x = 2 xy - z 2 , ∂φ y = 2 yz + x 2 , ∂φ z = y 2 - 2 zx . Therefore, φ( x , y , z ) = Z ( 2 xy - z 2 ) dx = x 2 y - xz 2 + C 1 ( y , z ) 2 yz + x 2 = ∂φ y = x 2 + C 1 y C 1 y = 2 yz C 1 ( y , z ) = y 2 z + C 2 ( z ) φ( x , y , z ) = x 2 y - xz 2 + y 2 z + C 2 ( z ) y 2 - 2 zx = ∂φ z
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Unformatted text preview: = -2 xz + y 2 + C 2 ( z ) ⇒ C 2 ( z ) = . Thus φ( x , y , z ) = x 2 y-xz 2 + y 2 z is a scalar potential for F , and F is conservative on 3 ....
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