Volume element | EPFL Graph Search

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form where the are the coordinates, so that the volume of any set can be computed by For example, in spherical coordinates , and so . The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the area element, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes 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 kind of measure on a manifold. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non-orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1-density. In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates In different coordinate systems of the form , , , the volume element changes by the Jacobian (determinant) of the coordinate change: For example, in spherical coordinates (mathematical convention) the Jacobian determinant is so that This can be seen as a special case of the fact that differential forms transform through a pullback as Consider the linear subspace of the n-dimensional Euclidean space Rn that is spanned by a collection of linearly independent vectors To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the is the square root of the determinant of the Gramian matrix of the : Any point p in the subspace can be given coordinates such that At a point p, if we form a small parallelepiped with sides , then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix This therefore defines the volume form in the linear subspace.

Buckling of pressurized spherical shells containing a through-thickness defect

Pedro Miguel Nunes Pereira de Almeida Reis, Matteo Pezzulla, Dong Yan

We present a study on the pressure buckling of thin, elastic spherical shells containing a thickness defect. Methodologically, we combine precision model experiments, finite element simulations, and a reduced axisymmetric shell model. We observe qualitativ ...

PERGAMON-ELSEVIER SCIENCE LTD2020

High Precision Capacitive Moisture Sensor for Polymers: Modeling and Experiments

Marco Mattavelli, Jean-Michel Sallese, Catherine Dehollain, Diego Ruben Barrettino

In this work a high precision capacitive moisture sensor for polymers was designed and fabricated. The device is part of a compact, fast, and economic moisture analyzer capable of reaching a similar precision than commercial moisture analyzers based on the ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2020