In three dimensional Cartesian coordinates the Hamilton operator, del, is written as
$\nabla= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix}$
The divergence of a vector field $A$ is written as
$\nabla \cdot A= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} =\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z} $
The curl is writtten as
$\nabla \times A= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \times \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} = \begin{pmatrix} \frac{\partial A_z}{\partial y} -\frac{\partial A_y}{\partial z} \\ \frac{\partial A_x}{\partial z} -\frac{\partial A_z}{\partial x} \\ \frac{\partial A_y}{\partial x} -\frac{\partial A_x}{\partial y} \end{pmatrix} $
I am trying to write these three equations for generalized coordinates using the Einstein summation convention and the basis vectors $e_1$, $e_2$, and $e_3$ for the scalars $x^1$, $x^2$, and $x^3$
So far I have the equations for gradient and divergence as
$\nabla= e_\mu \frac{\partial}{\partial x^\mu}$
$\nabla \cdot A= e_\mu \cdot\frac{\partial A_\mu}{\partial x^\mu}$
Are these equations correct? What would the equation for curl be?
$\nabla_aA_b = \partial_aA_b-\Gamma^c_{\space \space ab}A_c$
$\nabla_aA^b = \partial_aA^b+\Gamma^b_{\space \space ac}A^c$
$(\nabla f)^a = g^{ab}\partial_b$f
$\nabla^2f=\nabla_a\nabla^af=g^{ab}(\partial_a\partial_b-\Gamma^c_{\space \space ab}\partial_c)f$
$\nabla \cdot F = \nabla_aF^a= \frac{1}{\sqrt{g}}\partial_a(\sqrt{g}F^a)$
$(\nabla \times F)^\mu =?\frac{1}{\sqrt{g}} \epsilon^{\mu\nu\lambda}\partial_{\nu}F_{\lambda}$