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Laplace equation

  1. Apr 26, 2008 #1
    1. The problem statement, all variables and given/known data
    Verify that the function u=1/(x^2 + y^2 + z^2)^2 is a solution of the 3-dimensional Laplace equation uxx+uyy+uzz=0



    3. The attempt at a solution
    I know how to solve the partial derivatives, so I know that uxx=uyy=uzz for this problem. How can their sum equal 0?
     
  2. jcsd
  3. Apr 26, 2008 #2
    are you sure its not ^3/2?
     
  4. Apr 26, 2008 #3

    Dick

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    You mean ^(1/2), yes? 1/r is the Green's function for the Laplace equation.
     
  5. Apr 27, 2008 #4
    Ah, yes, the function should read:
    u=1/(x^2 + y^2 + z^2)^(1/2)

    Can you explain how the sum of the partial derivatives should equal zero, if their individual expressions are equal and positive?
     
  6. Apr 27, 2008 #5

    Dick

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    You said you know you to find the second derivatives. Then do it. The individual expression aren't 'equal and positive'. Tell me what is the second derivative u_xx? It has two terms which cancel when summed over x,y and z.
     
  7. Apr 27, 2008 #6
    For u_xx I'm getting 3(x^2 + y^2 + z^2)^-(5/2)
     
  8. Apr 27, 2008 #7

    Dick

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    That's not what I get. I get an 'x' in the numerator after the first derivative coming from the chain rule. When I apply the quotient rule to that to get the second derivative I get two terms.
     
  9. Apr 27, 2008 #8
    Ok, for my u_xx I now get -(x^2 + y^2 + z^2)^(-3/2) + 3(x^2) (x^2 + y^2 + z^2)^(-5/2)
     
  10. Apr 27, 2008 #9

    Dick

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    Right. Now sum over x, y and z.
     
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