scholarly journals Constructing p, n-forms from p-forms via the Hodge star operator and the exterior derivative

2020 ◽  
Vol 72 (6) ◽  
pp. 065402
Author(s):  
Jun-Jin Peng
Keyword(s):  
Exterior Derivative ◽  
Hodge Star Operator

scholarly journals Electromagnetic Wave Equation on Differential Form Representation

2018 ◽  
pp. 7-16
Author(s):  
I Gusti Ngurah Yudi Handayana
Keyword(s):  
Wave Equation ◽  
Electromagnetic Wave ◽  
Differential Form ◽  
Theoretical Physics ◽  
Exterior Derivative ◽  
Vector Representation ◽  
Minkowski Space Time ◽  
Form Representation ◽  
De Rham ◽  
Hodge Star Operator

One of the indispensable part of the theoretical physics interest is geometry differential. This one interest of physical area has been developed such as in electromagnetism. Maxwell's equations have been generalized in two covariant forms in differential form representation. A beautiful calculus vector in this representation, such as exterior derivative and Hodge star operator, lead this study. Electromagnetic wave equation has been expressed in differential form representation using Laplace-de Rham operator. Explicitly, wave equation shows the same form in Minkowski space-time like vector representation. This study is able to introduce us to learn application of differential form in physics.

scholarly journals $$H^1$$-conforming finite element cochain complexes and commuting quasi-interpolation operators on Cartesian meshes

CALCOLO ◽  
2021 ◽  
Vol 58 (2) ◽  
Author(s):  
Francesca Bonizzoni ◽  
Guido Kanschat
Keyword(s):  
Finite Element ◽  
Tensor Product ◽  
Arbitrary Order ◽  
Inner Product ◽  
Cochain Complex ◽  
Exterior Derivative ◽  
Product Construction ◽  
Cartesian Meshes ◽  
Conforming Finite Element ◽  
Interpolation Operators

AbstractA finite element cochain complex on Cartesian meshes of any dimension based on the $$H^1$$ H 1 -inner product is introduced. It yields $$H^1$$ H 1 -conforming finite element spaces with exterior derivatives in $$H^1$$ H 1 . We use a tensor product construction to obtain $$L^2$$ L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order.

The exterior derivative as a Killing vector field

1996 ◽  
Vol 93 (1) ◽  
pp. 157-170 ◽  
Author(s):  
J. Monterde ◽  
O. A. Sánchez-Valenzuela
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Exterior Derivative

scholarly journals The Hodge star operator on Schubert forms

Topology ◽  
2002 ◽  
Vol 41 (5) ◽  
pp. 945-960
Author(s):  
Klaus Künnemann ◽  
Harry Tamvakis
Keyword(s):  
Hodge Star Operator

scholarly journals CARTAN CALCULUS VIA PAULI MATRICES

2003 ◽  
Vol 18 (28) ◽  
pp. 5231-5259
Author(s):  
D. MAURO
Keyword(s):  
Vector Field ◽  
Path Integral ◽  
Classical Mechanics ◽  
Tensor Products ◽  
Lie Derivative ◽  
Exterior Derivative ◽  
Integral Formalism ◽  
Jacobi Fields ◽  
Finite Dimensional ◽  
Pauli Matrices

In this paper we will provide a new operatorial counterpart of the path-integral formalism of classical mechanics developed in recent years. We call it new because the Jacobi fields and forms will be realized via finite dimensional matrices. As a byproduct of this we will prove that all the operations of the Cartan calculus, such as the exterior derivative, the interior contraction with a vector field, the Lie derivative and so on, can be realized by means of suitable tensor products of Pauli and identity matrices.

scholarly journals A FERMIONIC HODGE STAR OPERATOR

1999 ◽  
Vol 14 (15) ◽  
pp. 965-976
Author(s):  
ALFRED DAVIS ◽  
TRISTAN HÜBSCH
Keyword(s):  
Field Theory ◽  
Wave Function ◽  
Bilinear Form ◽  
Operator Representation ◽  
Exterior Calculus ◽  
Star Operation ◽  
Hodge Star Operator

A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in space–times of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field theory, and induces a metric in the space of wave function(al)s just as in exterior calculus. If made real (hermitian), this induced metric turns out to be identical to the standard one constructed using hermitian conjugation; the utility of the induced complex bilinear form remains unclear.

A CONSTRUCTION OF SPACES OF COMPATIBLE DIFFERENTIAL FORMS ON CELLULAR COMPLEXES

2008 ◽  
Vol 18 (05) ◽  
pp. 739-757 ◽  
Author(s):  
SNORRE H. CHRISTIANSEN
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Differential Forms ◽  
Cochain Complex ◽  
Two Dimensional ◽  
Exterior Derivative ◽  
Whitney Forms ◽  
Reconstruction Operator ◽  
Mimetic Finite Differences ◽  
Cellular Complex

Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.

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