Volume form

Summary

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the line bundle \textstyle{\bigwedge}^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volu

Official source

https://en.wikipedia.org/wiki/Volume_form

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (3)

Related publications

Related people

Related units

Related concepts (40)

Related concepts

Related courses (5)

Related courses

Related lectures (10)

Related lectures

The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations

Given a principal bundle G hooked right arrow P -> B (each being compact, connected and oriented) and a G-invariant metric h(P) on P which induces a volume form mu(P), we consider the group of all unimodular automorphisms SAut(P, mu(P)) := {phi is an element of Diff(P) vertical bar phi*mu(P) = mu(P) and phi is G-equivariant) of P, and determines its Euler equation a la Arnold. The resulting equations turn out to be (a particular case of) the Euler-Yang-Mills equations of an incompressible classical charged ideal fluid moving on B. It is also shown that the group SAut(P, mu(P)) is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau(P) of P. (C) 2010 Elsevier B.V. All rights reserved.

2010

In this thesis, we study some linear and nonlinear problems involving differential forms. We begin by studying the problem of pullbacks which asks the following question: for two given differential forms, if one is the pullback of the other via a diffeomorphism satisfying some given condition. For volume forms, this problem was studied by Dacorogna-Moser giving a necessary and sufficient condition for the existence of the diffeomorphism with precise regularity. Our goal is to extend this result for general k-forms. We have obtained some necessary and sufficient conditions for two-forms and for some special classes of k-forms with sharp regularity. Then we turn our attention to the problem of differential inclusions involving differential forms. Although for zero-forms, the problem has been extensively studied, essentially nothing was known for higher forms including the curl operator. In this direction, we have obtained some necessary and some sufficient conditions for general k-forms unifying the study of the different cases. Moreover, we show that these necessary and sufficient conditions coincide for k = 1, solving the case of curl operator fairly completely. Besides these problems, we have studied some domain invariance property of the weighted-homogenous and non-homogenous Hardy constants as well. We have showed that the Hardy constant corresponding to these classes of inequalities enjoy, to some extent, the same domain invariance property as that of the Hardy constant corresponding to the standard Hardy's inequality.

EPFL2007

L'équation de rappel

Olivier Kneuss

In this thesis we are interested in the following problem : given two differential k–forms g and f, most of the time they will be assumed closed, on what conditions can we pullback g to f by a map φ ? In other words we ask when it is possible to solve the pullback equation φ*(g) = f. This equation originates from differential geometry. This work is divided into five main parts. In Part I we focus on exterior forms. The study of these forms, which is part of the domain of multilinear algebra, will allow us to understand the pullback equation at the algebraic level and will turn out to be a main first step for the study of the problem. Part II deals with the study of differential forms. A special attention will be brought to different versions of Poincaré lemma which will be constantly used all the way through this thesis. In Part III we solve the problem for volume forms k = n. This part, which will use very seldom the machinery of differential geometry, can be read independently of the previous two parts. Part IV sheds light on the case 0 ≤ k ≤ n – 1. Emphasis will be given on the case k = 2. The best results of this present work work will concern 2–forms and volume forms. The cases k = 0, 1, n – 1 are the easiest and will be discussed briefly while for the case 3 ≤ k ≤ n – 2, which will turn much more difficult even already at the algebraic level, we will only obtain results for k-forms with a special structure. Finally, in Part V, we gather numerous useful properties of Hödler spaces. These spaces will be very suitable for the pullback equation ; indeed there are an algebra (contrary to Sobolev spaces in general) and allow to get results with sharp regularity.

EPFL2011