Note: This questions was originally asked in iMechanica. The main confusion appears to be on whether Christoffel symbols should appear in the divergence of a field expressed in curvilinear coordinates or not.
As is known to all, the Laplace operator in cylindrical coordinates is defined as $$ \nabla^{2}=\dfrac{\partial^{2}}{\partial r^{2}}+\dfrac{1}{r}\dfrac{\partial}{\partial r}+\dfrac{1}{r^{2}}\dfrac{\partial^{2}}{\partial\phi^{2}}+\dfrac{\partial^{2}}{\partial z^{2}} $$ From here we can see that the $\nabla$ operator has the form $$ \nabla=\left(\dfrac{\partial}{\partial\bar{x}_{1}},\dfrac{\partial}{\partial\bar{x}_{2}},\dfrac{\partial}{\partial\bar{x}_{3}}\right)^{T} $$ rather than $$ \nabla=\left(\dfrac{\partial}{\partial r},\dfrac{\partial}{\partial\phi},\dfrac{\partial}{\partial z}\right)^{T}. $$ In the recent book ''Introduction to Metamaterials and Waves in Composites'' by Biswajit Banerjee, there are Christoffel symbols in the definition of divergence of tensors (see page 9), which implies the $\nabla$ is defined by $$ \nabla=(\dfrac{\partial}{\partial\bar{x}_{1}},\dfrac{\partial}{\partial\bar{x}_{2}},\dfrac{\partial}{\partial\bar{x}_{3}})^{T} $$
However, in the book ''Tensor Analysis'' (in Chinese, 2ed) by K. C. Huang et.al, they define $$ \nabla=\dfrac{\partial}{\partial x^{i}}\boldsymbol{g}^{i} $$ (see Eq.(4.2.14) in the book), where $x^{i}$ are the curvilinear coordinates. Hence Christoffel symbols do not occur in definition of divergence of a tensor. (see Eq.(4.4.2;4.4.3)).
Obviously, the two definitions for $\nabla$ are by no means compatible. So what is wrong?