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Unformatted text preview: ˆ‚ f + âˆ‚ f = 0 on D . Prove that
âˆ‚x2
âˆ‚y 2
âˆ‚D âˆ‚f
âˆ‚f
dx âˆ’
dy = 0.
âˆ‚y
âˆ‚x Solution: Greenâ€™s theorem states that
P dx + Q dy =
âˆ‚ D+ Let P = âˆ‚f
âˆ‚y D âˆ‚ Q âˆ‚P
âˆ’
âˆ‚x
âˆ‚y dx dy. and Q = âˆ’ âˆ‚f . Then
âˆ‚x
âˆ‚f
âˆ‚f
dx âˆ’
dy =
âˆ‚y
âˆ‚x
=âˆ’
=âˆ’ âˆ‚ D+ âˆ‚f
âˆ‚ âˆ‚f
âˆ‚
âˆ’
âˆ’
âˆ‚x
âˆ‚x
âˆ‚y âˆ‚y
2
2
âˆ‚f
âˆ‚f
+ 2 dx dy
2
âˆ‚x
âˆ‚y
D
D D 0 dx dy , dx dy since f is harmonic =0
Since this is zero then âˆ‚f
dx
âˆ‚ D âˆ‚y âˆ’ âˆ‚f
dy
âˆ‚x = 0 whichever direction we go around âˆ‚D ....
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This note was uploaded on 01/19/2014 for the course MATH 3010 taught by Professor Magpantay during the Winter '13 term at York University.
 Winter '13
 Magpantay
 Math, Vector Calculus

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