Understanding wedge products for differential forms

Question

I am trying to understand the derivation of coordinate expression for the Laplace-Beltrami operator (wiki here). The Wikipedia page says that $\nabla\cdot X$ is an operator mapping a function to a function. This would mean that $\text{vol}_n=\sqrt{\vert g\vert}dx^{1}\wedge\dots\wedge dx^n$ is a function on the manifold. How is this function defined?

Answer 1

$\DeclareMathOperator{\Vol}{vol}$In the page you link, $\Vol_{n}$ isn't a function, but a non-vanishing differential form of top degree, $n$.

The Lie derivative $L_{X}(\Vol_{n})$ of the volume form with respect to the vector field $X$ is also a form of top degree, hence a scalar function multiple of $\Vol{n}$. (At each point of an $n$-manifold, the space of $n$-forms is one-dimensional.) This scalar function is defined to be the divergence $\nabla \cdot X$ of $X$, i.e., $$ (\nabla \cdot X) \Vol_{n} = L_{X}(\Vol_{n}). $$