Prove Laplace Operator Unchanged in 3D Rotation

[erica1451]

Homework Statement

Prove that the Laplacian operator in dimension 3 is unchanged if the coordinates are rotated.

Homework 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

The Attempt at a Solution

I have no idea how to even start this problem

 

[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.