I'm trying to evaluate the Laplacian of a vector $\vec\omega$ using finite difference methods. I am doing this within a Cartesian grid where $\Delta_y<\Delta_x=\Delta_z$ and I want to know if I can use the traditional finite difference equations for a second derivative or if I need to derive my own.
If I just consider one component $\omega_x$ and take the Laplacian of that I think I should get the following. $$\nabla^2\omega_x|_{i,j,k}=\frac{\omega_x^{i+1,j,k}-2\omega_x^{i,j,k}+\omega_x^{i-1,j,k}}{\Delta_x^2}+\frac{\omega_x^{i,j+1,k}-2\omega_x^{i,j,k}+\omega_x^{i,j-1,k}}{\Delta_y^2}+\frac{\omega_x^{i,j,k+1}-2\omega_x^{i,j,k}+\omega_x^{i,j,k-1}}{\Delta_z^2}+O(\Delta_x^2)+O(\Delta_y^2)+O(\Delta_z^2)$$ Is this still correct when $\Delta_y<\Delta_x=\Delta_z$?