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
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
The volume element is then given by
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 .
Integration with volume shapes
If a volume form is on a manifold and an integrable function, then is the integral
defined via local maps as follows: Let local coordinates be such that
is positively oriented. Then you can in the map area as
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.
literature
- Volume form . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).