Supplementary mathematics/Volume element

In mathematics and calculus and geometry, a volume element generally provides a means to integrate a function according to its position in the volume of different coordinate systems such as spherical coordinates and cylindrical coordinates. Therefore, a volume element is an expression of the form:

$\mathrm {d} V=\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}$

where the $u_{i}$ are the coordinates, so that the volume of any set $B$ can be computed by:$\operatorname {Volume} (B)=\int _{B}\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}.$For example, in spherical coordinates $\mathrm {d} V=u_{1}^{2}\sin u_{2}\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}$, and so $\rho =u_{1}^{2}\sin u_{2}$.

The concept and rule of the volume element is not limited to the spatial coordinate system or three dimensions: in two dimensions, it is also known as a topic called the area element, and in this setting it is useful for performing tasks such as surface integrals. Under the change of coordinates, the volume element is changed by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a type of measure in a manifold. In an orientable differentiable manifold, a volume element usually arises from a volume form: the higher-order differential form. In a non-orientable manifold, the volume element is usually the absolute value of the (locally defined) volume form: it defines a density of a (locally defined) volume form: it defines a 1-density.