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Math Help - Harmonic Functions (Partial Derivatives)

  1. #1
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    Harmonic Functions (Partial Derivatives)

    The Laplace operator \Delta is defined by \Delta f = f_{xx} + f_{yy}. A function u(x,y) satisfying the Laplace equation \Delta u = 0 is called harmonic.
    Show that u(x,y) = x is harmonic.

    First thing, though: What does it mean for a function to be harmonic? The explanation they give is confusing to me. Can you show me how to do this so I can understand how to do my other problems related to this?
    Last edited by ZeroVector; October 26th 2010 at 10:38 PM.
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  2. #2
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    Suppose f(x,y) = 2x^2 - 2y^2. (This is just an example.)

    Then f_{xx} = 4 and f_{yy} = -4, which means \Delta f = f_{xx} + f_{yy} = 4 - 4 = 0. Therefore, f(x,y) is harmonic. If this value had been nonzero, then f(x,y) would not be harmonic.

    Does this make sense?
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  3. #3
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    You are told that \nabla f= f_{xx}+ f_{yy} and you were told that u being "harmonic" means that u satisfies \nabla u= 0. Putting those together, u(x,y) is "harmonic" if and only if u_{xx}+ u_{yy}= 0. Is that true for u(x,y)= x?
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