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.

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.

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 ONE-LOOP EFFECTIVE POTENTIAL FOR SCALAR AND VECTOR FIELDS ON HIGHER-DIMENSIONAL NONCOMMUTATIVE FLAT MANIFOLDS

2001 ◽  
Vol 16 (23) ◽  
pp. 1479-1486 ◽  
Author(s):  
A. A. BYTSENKO ◽  
A. E. GONÇALVES ◽  
S. ZERBINI
Keyword(s):  
Zeta Function ◽  
Vector Fields ◽  
Dimensional Regularization ◽  
Massless Scalar ◽  
Quantum Field Theories ◽  
Dimensional Manifold ◽  
Logarithmic Term ◽  
Scalar And Vector Fields ◽  
The One ◽  
Flat Manifolds

The non-planar contribution to the effective potentials for massless scalar and vector quantum field theories on D-dimensional manifold with p compact noncommutative extra dimensions is evaluated by means of dimensional regularization implemented by zeta function techniques. It is found that, the zeta function associated with the one-loop operator may not be regular at the origin. Thus, the related heat kernel trace has a logarithmic term in the short t asymptotic expansion. Consequences of this fact are briefly discussed.

scholarly journals Full in-plane strain tensor analysis using the microscale ring-core FIB milling and DIC approach

2016 ◽  
Vol 94 ◽  
pp. 47-67 ◽  
Author(s):  
Alexander J.G. Lunt ◽  
Enrico Salvati ◽  
Lifeng Ma ◽  
Igor P. Dolbyna ◽  
Tee K. Neo ◽  
...  
Keyword(s):  
Plane Strain ◽  
Strain Tensor ◽  
Tensor Analysis ◽  
Ring Core

scholarly journals Diffusion of Particles, Heat and Magnetic Fields in Compressible Turbulent Media

1991 ◽  
Vol 130 ◽  
pp. 71-74
Author(s):  
A.Z. Dolginov ◽  
N.A. Silant’ev
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Vector Fields ◽  
Particle Number ◽  
Particle Number Density ◽  
Turbulent Diffusivity ◽  
Number Density ◽  
Scalar And Vector Fields ◽  
Kinetic Coefficients ◽  
The Magnetic Field

AbstractA new method for the calculation of kinetic coefficients is presented. This method allows us to obtain the distribution of scalar and vector fields (such as the temperature, the admixture particle number density and the magnetic field) in turbulent cosmic media with any value of S = u0т0/R0. The explicit expression for the “turbulent” diffusivity DT is obtained. In some cases DT becomes negative, implying the clustering of the admixture particles in patches (a local increase of the temperature and magnetic fields). The magnetic α-effect is considered for the case S ~ 1.

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