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Thread: Harmonic Functions (Partial Derivatives)

  1. #1
    Oct 2010

    Harmonic Functions (Partial Derivatives)

    The Laplace operator $\displaystyle \Delta$ is defined by $\displaystyle \Delta f = f_{xx} + f_{yy}$. A function $\displaystyle u(x,y)$ satisfying the Laplace equation $\displaystyle \Delta u = 0$ is called harmonic.
    Show that $\displaystyle u(x,y) = x$ 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; Oct 26th 2010 at 10:38 PM.
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  2. #2
    Senior Member
    Jan 2010
    Suppose $\displaystyle f(x,y) = 2x^2 - 2y^2$. (This is just an example.)

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

    Does this make sense?
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  3. #3
    MHF Contributor

    Apr 2005
    You are told that $\displaystyle \nabla f= f_{xx}+ f_{yy}$ and you were told that u being "harmonic" means that u satisfies $\displaystyle \nabla u= 0$. Putting those together, u(x,y) is "harmonic" if and only if $\displaystyle u_{xx}+ u_{yy}= 0$. Is that true for u(x,y)= x?
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