I am learning Laplace–Beltrami operator for computer graphics, and I cannot understand the meaning of the square root of determinant of metric tensor in the defination?$\operatorname {vol} _{n}:={\sqrt {|g|}}\;dx^{1}\wedge \cdots \wedge dx^{n}$
I have already learned something about the ordinary Laplace operator and basic knowledge of differential form.
As far as I know, the metric tensor saves the information about the inner product of two vectors on the actual surface instead of the Euclidean space.
But I don't know how the Laplace–Beltrami operator extends the Laplace operator and why the metric tensor is used in this way.
Any help is appreciated.