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Laplace Operator

  1. Apr 27, 2007 #1
    1. The problem statement, all variables and given/known data
    Prove that the Laplacian operator in dimension 3 is unchanged if the coordinates are rotated.


    2. Relevant equations
    If S is a rotation (S*S=I) and if x'=Sx then show d^2/dx1'^2 + d^2/dx2'^2 +d^2/dx3'^2 = d^2/dx1^2 + d^2/dx2^2 + d^2/dx3^2


    3. The attempt at a solution
    I have no idea how to even start this problem :confused:
     
  2. jcsd
  3. Apr 27, 2007 #2

    cristo

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    Staff Emeritus
    Science Advisor

    Well, can you write an expression for the matrix S such that S is a rotation, and the new coordinates are defined as x'=Sx where x=(x,y,z). Try writing the matrix for a rotation about the z axis, say, first. Then you need to prove that the equation you give holds for the primed coordinates.

    If you manage to show this, you can argue by symmetry for rotations about the x and y axes.
     
    Last edited: Apr 27, 2007
  4. Apr 29, 2007 #3
    I understand how to do the problem now. Thanks for your help.
     
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