Confusing variable change calculation

[AxiomOfChoice]

Suppose you start with a system of N particles identified by position vectors $r_1, r_2, \ldots, r_N$ and masses $\mu_1, \mu_2, \ldots, \mu_N$. Then (quantum mechanically) the kinetic energy operator for this system is given by (assuming $\hbar = 1$)

$$T = \sum_{i = 1}^N -\frac{\Delta_i}{2\mu_i},$$

where $\Delta_i = \Delta_{r_i}$ is the Laplacian in the variable $r_i$. Now, let's make the change of variables $\eta_i = r_i - r_N$. According to the book I have, the kinetic energy in the new variables looks like

$$T' = \sum_{i = 1}^{N-1} -\frac{\Delta_i}{2\mu_i'} + \sum_{i < j} \nabla_i \cdot \nabla_j,$$

where $\Delta_i = \Delta_{\eta_i}$, $\nabla_i = \nabla_{\eta_i}$, and

$$\frac{1}{\mu'_i} = \frac{1}{\mu_i} + \frac{1}{\mu_N}.$$

I simply do not see how this is true. It seems that, if one applied the chain rule, one obtains $\Delta_{\eta_i} = \Delta_{r_i}$; is that not true?

 

[fresh_42]

No. It depends on the connection we use, in this case probably Levi-Civita, because the coordinate transformation changes the evaluation point of the gradient field.