Integration on Manifolds

The integral of a scalar function f(x) over a region V in cartesian coordinates is

${\int }_V{d}^{d}xf(x)$

and under a coordinate transformation,

$d^{d}x=J{d}^d\tilde{x}$

where J is theJacobian

$J=\left|\begin{array}{c}\frac{\partial x^{\mu }}{\partial {\tilde{x}}^{\nu }}\end{array}\right|$

Similarly, for the metric,

$\tilde{g}=\left|\begin{array}{c}{\tilde{g}}_{\mu \nu }\end{array}\right|=-J^2⟹J=\sqrt{-\tilde{g}}$

Thus, in a general coordinate system, the integral of a function f over a region V is defined as

${\int }_V{d}^{d}x\sqrt{-g}f(x)$

In particular, if we integrate with f(x)=1, the result is the proper volume of the region V.