Vector Field Theory

1985 ◽  
pp. 454-495
Author(s):  
Murray H. Protter ◽  
Charles B. Morrey
Keyword(s):  
Field Theory ◽  
Vector Field

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.

VECTOR FIELD THEORY

1980 ◽  
pp. 1081-1092
Author(s):  
I.S. GRADSHTEYN ◽  
I.M. RYZHIK
Keyword(s):  
Field Theory ◽  
Vector Field

scholarly journals A new approach to anomalous axial vector field theory – II

2021 ◽  
pp. 2150155
Author(s):  
A. K. Kapoor
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vector Field ◽  
Axial Vector ◽  
Quantum Field ◽  
Stochastic Quantization ◽  
New Approach ◽  
Four Dimensions

This work is continuation of a stochastic quantization program reported earlier. In this paper, we propose a consistent scheme of doing computations directly in four dimensions using conventional quantum field theory methods.

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 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.

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