Some old and new techniques for the practical use of exterior differential forms

1976 ◽  
pp. 136-161 ◽  
Author(s):  
Frank B. Estabrook
Keyword(s):  
Differential Forms ◽  
New Techniques ◽  
Exterior Differential

Exterior Differential Forms

2013 ◽  
pp. 219-326
Author(s):  
Erdoğan S. Şuhubi
Keyword(s):  
Differential Forms ◽  
Exterior Differential

scholarly journals POINCARÉ AND SOBOLEV INEQUALITIES FOR DIFFERENTIAL FORMS IN HEISENBERG GROUPS AND CONTACT MANIFOLDS

2020 ◽  
pp. 1-52
Author(s):  
ANNALISA BALDI ◽  
BRUNO FRANCHI ◽  
PIERRE PANSU
Keyword(s):  
Differential Form ◽  
Scale Invariance ◽  
Differential Forms ◽  
Sobolev Inequalities ◽  
Exact Form ◽  
Heisenberg Groups ◽  
Contact Manifolds ◽  
Bounded Geometry ◽  
Exterior Differential ◽  
Rumin Complex

Abstract In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$ , where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$ , which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$ . In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

scholarly journals Non-Riemannian description of Robinson–Trautman spacetimes in Brans–Dicke theory of gravity

2019 ◽  
Vol 28 (04) ◽  
pp. 1950070
Author(s):  
Muzaffer Adak ◽  
Tekin Dereli ◽  
Yorgo Şenikoğlu
Keyword(s):  
Scalar Field ◽  
Differential Forms ◽  
Scalar Tensor Theory ◽  
Field Equations ◽  
Exterior Differential ◽  
Tensor Theory ◽  
Theory Of Gravitation ◽  
Theory Of Gravity ◽  
Future Reference

The variational field equations of Brans–Dicke scalar-tensor theory of gravitation are given in a non-Riemannian setting in the language of exterior differential forms over four-dimensional spacetimes. A conformally rescaled Robinson–Trautman metric together with the Brans–Dicke scalar field are used to characterize algebraically special Robinson–Trautman spacetimes. All the relevant tensors are worked out in a complex null basis and given explicitly in an appendix for future reference. Some special families of solutions are also given and discussed.

On the Vector-Valued Cauchy Stress 2-Form and Its Stress Rate

2007 ◽  
Author(s):  
H. Murakami
Keyword(s):  
Shear Stress ◽  
Differential Forms ◽  
Shear Loading ◽  
Stress Rate ◽  
Curvilinear Coordinates ◽  
Lie Derivative ◽  
Cauchy Stress ◽  
Exterior Differential ◽  
Correct Definition ◽  
Vector Valued

Using exterior differential forms, basic equations of continuum mechanics are presented in direct notation. To this end, Elie Cartan’s vector-valued Cauchy stress 2-form is introduced. Its Lie derivative along the world line becomes the Truesdell stress rate. In the presentation, the notation adopted by Theodore Frankel (The Geometry of Physics, Cambridge, New York, 1997) is utilized. With the use of exterior differential forms, complicated computations in tensor analyses in curvilinear coordinates are dramatically simplified. As specific examples, the following subjects are presented: (i) Piola transformations of the Cauchy stress 2-form and (ii) simple shear deformation using the Lie derivative of the Cauchy stress 2-form, i.e., the Truesdell stress rate. It is known that under monotonic shear loading, if inappropriate stress-rates are used, shear stress oscillates. With the use of geometrically correct stress-rate, the shear stress monotonically increases. Thereby, the search for an appropriate stress rate reduces to the correct definition of the stress 2-form and the computation of its Lie derivative with respect to velocity.

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