I learned the proof of the fact that 3-D laplacian is invariant under all rigid motions in space in Strauss's Partial Differential Equations:
Any rotation in three dimensions is given by ${\bf x'}=B{\bf x},$ where $B$ is an orthogonal matrix. The laplacian is $\Delta u=\sum_{i=1}^3 u_{ii}=\sum_{i,j=1}^3\delta_{ij}u_{ij}$ where the subscripts on $u$ denote partial derivatives. Therefore, $ \Delta u=\sum_{k,l}\Big(\sum_{i,j}b_{ki}\delta_{ij}b_{lj}\Big)u_{k'l'} =\sum_{k,l}\delta_{kl}u_{k'l'} =\sum_{k}u_{k'k'}. $
Added: I believe the chain rule is needs here, but I don't see how it is applied here.
Here is my question:
How can I get the first equality, i.e. $\Delta u=\sum_{k,l}\Big(\sum_{i,j}b_{ki}\delta_{ij}b_{lj}\Big)u_{k'l'}?$