Volume shape

A volume shape is a mathematical object that is required for integration over spatial areas, especially when using special coordinate systems , i.e. a special case of a volume .

In physics and engineering, terms such as infinitesimal volume element or measure factor are also used .

Examples in 3 dimensions

• Cartesian coordinates : ${\displaystyle \mathrm {d} V=\mathrm {d} x\cdot \ \mathrm {d} y\cdot \ \mathrm {d} z}$
• Cylindrical coordinates : ${\displaystyle \mathrm {d} V=\rho \cdot \ \mathrm {d} \rho \cdot \ \mathrm {d} \varphi \cdot \ \mathrm {d} z}$
• Spherical coordinates : ${\displaystyle \mathrm {d} V=\ r^{2}\cdot \sin \theta \cdot \ \mathrm {d} r\cdot \ \mathrm {d} \theta \cdot \ \mathrm {d} \varphi }$

Calculation in 3 dimensions

The volume element in three dimensions can be calculated according to the transformation theorem with the help of the functional determinant . The Jacobian matrix for the transformation from the coordinates to is defined here by ${\displaystyle \det J}$ ${\displaystyle \{x_{1},x_{2},x_{3}\}}$ ${\displaystyle \{x'_{1},x'_{2},x'_{3}\}}$

${\displaystyle J={\frac {\partial (x_{1},x_{2},x_{3})}{\partial (x'_{1},x'_{2},x'_{3})}}.}$

The volume element is then given by

${\displaystyle \mathrm {d} V'=|\det J|\,\mathrm {d} x'_{1}\,\mathrm {d} x'_{2}\,\mathrm {d} x'_{3}.}$

Mathematical definition

From a mathematical point of view, a volume form on a -dimensional manifold is a nowhere vanishing differential form of degree . In the case of an oriented Riemannian manifold , a canonical volume form results from the metric used, which assumes the value 1 on a positively oriented orthonormal basis . This is called the Riemann volume shape . ${\displaystyle n}$ ${\displaystyle n}$

Integration with volume shapes

If a volume form is on a manifold and an integrable function, then is the integral ${\displaystyle \omega }$ ${\displaystyle M}$ ${\displaystyle f}$

${\displaystyle \int _{M}f\cdot \omega }$

defined via local maps as follows: Let local coordinates be such that ${\displaystyle x_{1},\ldots ,x_{n}}$

${\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}}$

is positively oriented. Then you can in the map area as ${\displaystyle f\cdot \omega }$

${\displaystyle g\cdot \mathrm {d} x_{1}\wedge \ldots \wedge \mathrm {d} x_{n}}$

write; the integral is then the ordinary Lebesgue integral of . For the integral over whole , a partition of one or a decomposition of the manifold into disjoint measurable subsets can be used. The transformation theorem shows that this definition is not card-dependent. ${\displaystyle g}$ ${\displaystyle M}$

literature

• Volume form . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).