Quiz 3 Solution

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Unformatted text preview: f + ∂ f = 0 on D . Prove that ∂x2 ∂y 2 ∂D ∂f ∂f dx − dy = 0. ∂y ∂x Solution: Green’s theorem states that P dx + Q dy = ∂ D+ Let P = ∂f ∂y D ∂ Q ∂P − ∂x ∂y dx dy. and Q = − ∂f . Then ∂x ∂f ∂f dx − dy = ∂y ∂x =− =− ∂ D+ ∂f ∂ ∂f ∂ − − ∂x ∂x ∂y ∂y 2 2 ∂f ∂f + 2 dx dy 2 ∂x ∂y D D D 0 dx dy , dx dy since f is harmonic =0 Since this is zero then ∂f dx ∂ D ∂y − ∂f dy ∂x = 0 whichever direction we go around ∂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.

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