Laplace Operator

[erica1451]

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:

[cristo]

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.

[erica1451]

I understand how to do the problem now. Thanks for your help.