15_2_14 - + y 2 ) = m ln | r | , as a scalar potential for...

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14. For v ( x , y ) = m ( x i + y j ) x 2 + y 2 , we have ∂v 1 y = - 2 mxy ( x 2 + y 2 ) 2 = ∂v 2 x , so v may be conservative, except at ( 0 , 0 ) . We have φ( x , y ) = m Z x dx x 2 + y 2 = m 2 ln ( x 2 + y 2 ) + C 1 ( y ) my x 2 + y 2 = ∂φ y = my x 2 + y 2 + dC 1 dy . Thus we may take C 1 ( y ) = 0, and obtain φ( x , y ) = m 2 ln ( x 2
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Unformatted text preview: + y 2 ) = m ln | r | , as a scalar potential for the velocity field v of a line source of strength of m ....
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