Abstract
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.
Highlights
Many different processes and phenomena are modeled using the mathematical apparatus of the field theory
If the field source is concentrated at a point, its differential characteristics are conveniently described in the spherical coordinates
The above expressions of the differential-geometrical characteristics of the scalar and vector fields, presented in normal conical coordinates, make it possible to design a geometrical model of the field and involve visual computer graphics in its study
Summary
Many different processes and phenomena are modeled using the mathematical apparatus of the field theory. The field theory with point and linear sources is most fully investigated and described in the scientific literature [3] This is because the describing mathematical apparatus is based on the use of spherical and cylindrical coordinate systems. When describing fields of complex structure, when the source has a shape different from a point or a straight line, mathematical complications arise They require new research methods in studying the processes occurring in the environment of such fields [4, 5]. When studying the level surfaces of thermal fields with a source in the form of external conical surfaces, it turned out that these surfaces do not coincide with families of equidistant cones They are not coordinate in any of the well-known coordinate systems. It is mandatory to preserve the condition of coincidence of coordinate families of surfaces with field level surfaces or with limiting surfaces of the process
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