scholarly journals Field theory in normal toroidal coordinates

2018 ◽  
Vol 193 ◽  
pp. 03022
Author(s):  
Dmitry V. Nesnov
Keyword(s):  
Field Theory ◽  
Vector Field ◽  
Computer Graphics ◽  
Geometric Modeling ◽  
Vector Fields ◽  
Scientific Paper ◽  
Curvilinear Coordinates ◽  
Coordinate Systems ◽  
Scalar And Vector Fields ◽  
Toroidal Coordinates

Field theory is widely represented in spherical and cylindrical coordinate systems, since the mathematical apparatus of these coordinate systems has been thoroughly studied. Sources of field with more complex structures require new approaches to their study. The purpose of this research is to adapt the field theory referred to curvilinear coordinates and represent it in normal toroidal coordinates. Another purpose is to develop the foundations of geometric modeling with the use of computer graphics for visualizing the level surfaces. The dependence of normal toroidal coordinates on rectangular Cartesian coordinates and Lame coefficients is shown in this scientific paper. Differential characteristics of scalar and vector fields in normal toroidal coordinates are obtained: scalar and vector field laplacians, divergence, and rotation of vector field. The example shows the technique of modeling the field and its further computer visualization. The technique of reading the internal equation of the surface is presented and the influence of the values of the parameters on the shape of the surface is shown. For the first time, expressions of scalar and vector field characteristics in normal toroidal coordinates are obtained, the fundamentals of geometric modeling of fields with the use of computer graphics tools are developed for the purpose of providing visibility for their study.

scholarly journals Field theory in normal conical coordinates

2021 ◽  
Vol 244 ◽  
pp. 09004
Author(s):  
Dmitry Nesnov
Keyword(s):  
Field Theory ◽  
Vector Field ◽  
Computer Graphics ◽  
Vector Fields ◽  
Complex Structure ◽  
Curvilinear Coordinates ◽  
Mathematical Apparatus ◽  
Spherical Coordinate ◽  
Geometrical Modeling ◽  
Scalar And Vector Fields

In the scientific literature, the field theory is most fully covered in the cylindrical and spherical coordinate systems. This is explained by the fact that the mathematical apparatus of these systems is most well studied. When the source of field has a more complex structure than a point or a straight line, there is a need for new approaches to their study. The goal of this research is to adapt the field theory related to curvilinear coordinates in order to represent it in the normal conical coordinates. In addition, an important part of the research is the development of a geometrical modeling apparatus for scalar and vector field level surfaces using computer graphics. The paper shows the dependences of normal conical coordinates on rectangular Cartesian coordinates, Lame coefficients. The differential characteristics of the scalar and vector fields in normal conical coordinates are obtained: Laplacian of scalar and vector fields, divergence, rotation of the vector field. The example case shows the features of the application of the mathematical apparatus of geometrical field modeling in normal conical coordinates. For the first time, expressions for the characteristics of the scalar and vector fields in normal conical coordinates are obtained. Methods for geometrical modeling of fields using computer graphics have been developed to provide illustration in their study.

scholarly journals Charge conservation, entropy current and gravitation

2021 ◽  
Author(s):  
Sinya Aoki ◽  
Tetsuya Onogi ◽  
Shuichi Yokoyama
Keyword(s):  
Field Theory ◽  
Vector Field ◽  
Vector Fields ◽  
Plane Waves ◽  
Momentum Tensor ◽  
Entropy Density ◽  
Global Symmetry ◽  
Timelike Vector ◽  
First Law Of Thermodynamics ◽  
Entropy Current

We propose a new class of vector fields to construct a conserved charge in a general field theory whose energy–momentum tensor is covariantly conserved. We show that there always exists such a vector field in a given field theory even without global symmetry. We also argue that the conserved current constructed from the (asymptotically) timelike vector field can be identified with the entropy current of the system. As a piece of evidence we show that the conserved charge defined therefrom satisfies the first law of thermodynamics for an isotropic system with a suitable definition of temperature. We apply our formulation to several gravitational systems such as the expanding universe, Schwarzschild and Banãdos, Teitelboim and Zanelli (BTZ) black holes, and gravitational plane waves. We confirm the conservation of the proposed entropy density under any homogeneous and isotropic expansion of the universe, the precise reproduction of the Bekenstein–Hawking entropy incorporating the first law of thermodynamics, and the existence of gravitational plane wave carrying no charge, respectively. We also comment on the energy conservation during gravitational collapse in simple models.

scholarly journals de Sitter symmetries and inflationary scalar-vector models

2016 ◽  
Vol 21 (3) ◽  
pp. 219 ◽  
Author(s):  
Cesar Alonso Valenzuela Toledo ◽  
Juan Beltrán Almeida ◽  
Josue Motoa-Manzano
Keyword(s):  
Field Theory ◽  
Euclidean Space ◽  
Correlation Functions ◽  
Vector Fields ◽  
Conformal Symmetry ◽  
De Sitter Space ◽  
Dual Theory ◽  
De Sitter ◽  
Curvature Perturbation ◽  
Scalar And Vector Fields

<div class="page" title="Page 1"><div class="section"><div class="layoutArea"><div class="column"><p><span>In this paper, we study the correspondence between a field theory in de Sitter space in D-dimensions and a dual conformal feld theory in a euclidean space in (D - 1)-dimensions. In particular, we investigate the form in which this correspondence is established for a system of interacting scalar and a vector fields propagating in de Sitter space. We analyze some necessary (but not sucient) conditions for which conformal symmetry is preserved in the dual theory in (D - 1)-dimensions, making possible the establishment of the correspondence. The discussion that we address in this paper is framed on the context of <em>inationary cosmology</em>. Thusly, the results obtained here pose some relevant possibilities of application to the calculation of the fields’s correlation functions and of the <em>primordial curvature perturbation</em> \zeta, in inationary models including coupled scalar and vector fields.</span></p></div></div></div></div>

Phase structure of the scalar Yukawa model with compactified spatial dimensions

2016 ◽  
Vol 31 (20) ◽  
pp. 1650121 ◽  
Author(s):  
L. M. Abreu ◽  
A. P. C. Malbouisson ◽  
E. S. Nery
Keyword(s):  
Field Theory ◽  
Phase Structure ◽  
Vector Fields ◽  
Boundary Effects ◽  
Quantum Field ◽  
Yukawa Model ◽  
Complex Scalar ◽  
Scalar And Vector Fields ◽  
Spatial Dimensions ◽  
Complex Scalar Field

In this work, we investigate the thermodynamic behavior of the generalized scalar Yukawa model, composed of a complex scalar field interacting with real scalar and vector fields. In particular, boundary effects on the phase structure are discussed using methods of quantum field theory on toroidal topologies. We concentrate on the dependence of the thermodynamics with the number of compactified spatial dimensions. In this sense, the phase transitions are analyzed and compared with the system in the situations of one, two and three compactified spatial dimensions. Our findings suggest that the presence of more boundaries tends to inhibit the broken phase.

scholarly journals Power-law Genesis: Strong coupling and Galileon-like vector fields

2020 ◽  
Vol 35 (37) ◽  
pp. 2050305
Author(s):  
P. K. Petrov
Keyword(s):  
Field Theory ◽  
Vector Field ◽  
Power Law ◽  
Vector Fields ◽  
Energy Condition ◽  
Classical Field Theory ◽  
Flat Space ◽  
Space Time ◽  
Classical Field ◽  
Negative Power

A simple way to construct models with early cosmological Genesis epoch is to employ bosonic fields whose Lagrangians transform homogeneously under scaling transformation. We show that in these theories, for a range of parameters defining the Lagrangian, there exists a homogeneous power-law solution in flat space-time, whose energy density vanishes, while pressure is negative (power-law Genesis). We find the condition for the legitimacy of the classical field theory description of such a situation. We note that this condition does not hold for our earlier Genesis model with vector field. We construct another model with vector field and power-law background solution in flat space-time, which is legitimately treated within classical field theory, violates the Null Energy Condition (NEC) and is stable. Upon turning on gravity, this model describes the early Genesis stage.

TENSOR ANALYSIS OF MAIN DIFFERENTIAL OPERATORS UTILIZED FOR DESCRIPTION OF OCEAN HYDRODYNAMICS IN CURVILINEAR ORTHOGONAL COORDINATE SYSTEM

2019 ◽  
Vol 47 (2) ◽  
pp. 139-171
Author(s):  
V.M. Kamenkovich ◽  
D.A. Nechaev
Keyword(s):  
Vector Fields ◽  
Differential Operators ◽  
Strain Tensor ◽  
Symmetric Tensor ◽  
Ocean Dynamics ◽  
Tensor Analysis ◽  
Coordinate Systems ◽  
Invariant Representation ◽  
Material Derivative ◽  
Scalar And Vector Fields

The paper presents an analysis of tensor expressions in different curvilinear orthogonal coordinate systems. The analysis reveals specific properties of a number of approximated coordinate systems widely used in the studies of ocean dynamics. The paper consists of two parts. The part 1 presents a brief overview of the key definitions and important relations of tensor analysis which are utilized in part 2 of the paper. The part 2 considers invariant representation of different types of vector products, divergence of vector field and divergence of symmetric tensor of rank 2, gradient of a scalar filed, curl of a vector filed. The part 2 also discusses specific properties of the rate-of-strain tensor, general form of the Laplace operator, properties of operator nabla, and general forms of material derivative for scalar and vector fields. The equations for the properties under consideration are derived for the physical components of the corresponding tensors.

Some aspects of the pedagogical model of constructive geometric modeling

2020 ◽  
Author(s):  
Denis Voloshinov ◽  
K. Solomonov ◽  
Lyudmila Mokretsova ◽  
Lyudmila Tishchuk
Keyword(s):  
Computer Graphics ◽  
Geometric Modeling ◽  
Educational Process ◽  
Educational Institutions ◽  
Software Systems ◽  
Teaching Methodology ◽  
Models Of Teaching ◽  
Software Product ◽  
The University

The application of constructive geometric modeling to pedagogical models of teaching graphic disciplines today is a promising direction for using computer technology in the educational process of educational institutions. The essence of the method of constructive geometric modeling is to represent any operation performed on geometric objects in the form of a transformation, as a result of which some constructive connection is established, and the transformation itself can be considered as a result of the action of an abstract cybernetic device. Constructive geometric modeling is a popular information tool for information processing in various applied areas, however, this tool cannot be appreciated without the presence of appropriate software systems and developed design techniques. Traditionally, constructive geometric modeling is used in the design of mechanical engineering, energy, aircraft and shipbuilding facilities, in architectural and design engineering. The need to study descriptive geometry at the university in recent years has something in common with the issues of mastering graphic packages of computer programs in the framework of the new discipline "Engineering and Computer Graphics". The well-known KOMPAS software product is considered the simplest and most attractive for training. It should be noted the important role of graphic packages in the teaching of geometric disciplines that require a figurative perception of the material by students. Against the background of a reduction in classroom hours, computer graphics packages are practically the only productive teaching methodology, successfully replacing traditional tools - chalk and blackboard.

scholarly journals Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehwish Bari ◽  
Ghulam Mustafa ◽  
Abdul Ghaffar ◽  
Kottakkaran Sooppy Nisar ◽  
Dumitru Baleanu
Keyword(s):  
Computer Graphics ◽  
Geometric Modeling ◽  
Tension Control ◽  
Free Form ◽  
Subdivision Surfaces ◽  
Smooth Curves ◽  
Practical Applications ◽  
Tension Parameter ◽  
Exact Reproduction ◽  
Free Form Surfaces

AbstractSubdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having $C^{1}$ C 1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.

A Multi-Parameter Integrated Magnetometer Based on Combination of Scalar and Vector fields

2021 ◽  
pp. 1-1
Author(s):  
Jian Ge ◽  
Rui Wang ◽  
Haobin Dong ◽  
Huan Liu ◽  
Qianwei Zheng ◽  
...  
Keyword(s):  
Vector Fields ◽  
Scalar And Vector Fields

scholarly journals Pairings between bounded divergence-measure vector fields and BV functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Virginia De Cicco ◽  
Annalisa Malusa
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Vector Fields ◽  
Hyperbolic Conservation Laws ◽  
Divergence Measure ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Bv Functions ◽  
Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

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