divergence in polar coordinates

$\DeclareMathOperator\div{div}$The formula for $\nabla\cdot X$ is incorrect. The notation with the 'usual' dot product is misleading. Properly it is: $$\div F = \frac 1\rho\frac{\partial(\rho F^i)}{\partial x^i}$$ where $\rho=\sqrt{\det g}$ is the coefficient of the differential volume element $dV=\rho\, dx^1\wedge\ldots \wedge dx^n$, meaning $\rho$ is also the Jacobian determinant, and where $F^i$ are the components of $F$ with respect to an unnormalized basis.

In polar coordinates we have $\rho=\sqrt{\det g}=r$, and: $$\div X = \frac 1r \frac{\partial(r X^r)}{\partial r} + \frac 1r\frac{\partial(r X^\theta)}{\partial \theta}$$

In the usual normalized coordinates $X=\hat X^{r}\frac{\partial}{\partial r} + \hat X^{\theta}\frac 1r\frac{\partial}{\partial\theta}$ this becomes: $$\div X = \frac 1r \frac{\partial(r \hat X^{r})}{\partial r} + \frac 1r\frac{\partial \hat X^{\theta}}{\partial \theta}$$ which agrees with the usual formula given in calculus books.